Cutoff phenomenon of the Glauber dynamics for the Ising model on complete multipartite graphs in the high temperature regime

Heejune Kim School of Mathematics, University of Minnesota. [email protected]

Abstract.

In this paper, the Glauber dynamics for the Ising model on the complete multipartite graph $K_{np_{1},\dots,np_{m}}$ is investigated where $0<p_{i}<1$ is the proportion of the vertices in the $i$ th component. We show that the dynamics exhibits the cutoff phenomena at $t_{n}\colonequals\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ with window size $O(n)$ in the high temperature regime $\beta<\beta_{cr}$ where $\beta_{cr}$ is a constant only depending on $p_{1},\dots,p_{m}$ . Exponentially slow mixing is shown in the low temperature regime $\beta>\beta_{cr}$ .

Key words and phrases:

Markov chains, Ising model, Mixing time, Cutoff, Coupling, Glauber dynamics, Heat-bath dynamics, Mean-field model

2020 Mathematics Subject Classification:

60J10, 60K35, 82C20

1. Introduction and preliminaries

Informally, the cutoff phenomenon is an abrupt transition of a Markov chain to its equilibrium when the system under consideration is sufficiently large (see Section 1.3 for a rigorous definition). To the author’s knowledge, the first rapid mixing result appeared in [4] on the symmetric group while considering random transpositions. Shortly afterward, [2] showed that the top-in-at-random card-shuffle precisely exhibits a cutoff phenomenon, initiating the whole industry of the cutoff phenomenon.

As pointed out in [12], only a few examples of cutoff were known regarding the Glauber dynamics of the Ising model (see Section 1.2 for formal definitions), such as that of \citesdinglevin on complete graphs and of \citesLubetzky_2012lubetzky2012cutofflubetzky2014 on lattices. Recent researches have mainly focused on lattices. A breakthrough paper by [12] showed cutoff with a continuous-time window $O(\ln\ln n)$ for this longstanding problem. An improvement on the window size to optimal $O(1)$ was made by the same authors in [14] with the information percolation framework. By the same technique, the authors illustrated the existence of cutoff in high enough temperatures for the Ising model of any sequence of graphs with a bounded degree in [15]. Mean-field Potts model on complete graphs was comprehensively explored in [3], again verifying the cutoff phenomenon in high temperatures. For the bipartite Potts model, [7] proved the cutoff phenomena in the high temperatures using their aggregate path coupling method.

The purpose of this paper is to investigate the Glauber dynamics for the Ising model on complete multipartite graphs. (Exact definitions are given in the rest of the introduction.) Indeed, we identify the critical temperature and establish cutoff in the high temperature regime. On the other hand, exponentially slow mixing is established in the low temperature regime. The significance of our setting is that complete multipartite graphs have an intermediate geometry between the complete graphs which have no geometry at all (e.g. [10]), and lattices which have a strong geometry (e.g. [12]). Thus, our result serves as a midway example between those two extreme cases. The method of proof hinges on generalizations of the tools in [10], notably the two-coordinate chain thereof.

Due to the nature of complete multipartite graphs, our model can be considered as a block spin Ising model with no interaction inside each block. Such mean-field block models naturally occur in statistical physics when modelling metamagnets (see [8]) and in studies on social interactions (see, e.g., [6]). A recent paper by [9] contains an excellent introduction to this line of work.

When it comes to cutoff phenomenon on finite graphs, it is easy to convert the discrete-time results to that of the continuous-time and vice versa. Hence, we only consider discrete-time chains.

1.1. Notations

Boldface letters are used to denote vectors or matrices. Inequalities between vectors and matrices are defined element-wise. The dependence of any quantities on the number of vertices $n$ is understood throughout the paper. Some important quantities not depending on $n$ will be explicitly mentioned. We will write $\mathbf{e}_{j}$ to be the $j$ th vector in the standard basis of $\mathbb{R}^{m}$ . The lower case $t$ will always denote time. Let $\circ$ denote the Hadamard product between matrices. More precisely, $B\circ C=(B_{ij}C_{ij})$ whenever $B=(B_{ij})$ and $C=(C_{ij})$ are matrices with the same dimensions.

1.2. Ising model and Glauber dynamics

Let $G=(V,E)$ be a finite graph with the vertex set $V$ and the edge set $E$ . Elements of $\Omega\colonequals\{\pm 1\}^{V}$ are called configurations. In the absence of external fields, the Ising model on $G$ is a distribution $\mu$ called the Gibbs distribution on $\Omega$ given by

\mu(\sigma)\colonequals\frac{e^{-\beta H(\sigma)}}{Z(\beta)}

where $\sigma\in\Omega$ , $\beta\geq 0$ , $H(\sigma)=-\sum_{ij\in E}h_{ij}\sigma(i)\sigma(j)$ , and $Z(\beta)$ is a normalizing factor. Assuming an isotropic interaction strength between the vertices, we set $h_{ij}=1/|V|$ . The physical interpretation of $H(\sigma)$ is the energy of the whole spin system with the configuration $\sigma$ . We call each $\sigma(v)$ the spin at site $v$ .

The Glauber dynamics for the Ising model is a reversible Markov chain with respect to the Gibbs distribution satisfying the following rule. At each time, choose a site uniformly at random in $V$ and update the spin at the chosen site according to $\mu$ conditioned on the set of configurations having the same spins at all the sites except the chosen one. The Glauber dynamics for the Gibbs distribution $\mu$ is irreducible, aperiodic, and reversible with $\mu$ as its unique stationary distribution. For the Ising model, it is easy to see that the probability of updating to $\pm 1$ at the chosen site $v$ is $r_{\pm}(S)$ where

\displaystyle r_{\pm}(x)\colonequals\frac{e^{\pm\beta x}}{e^{\beta x}+e^{-\beta x}}=\frac{1\pm\tanh(\beta x)}{2};\quad x\in\mathbb{R}

(1)

and $S=\sum_{vv^{\prime}\in E}\sigma(v^{\prime})/|V|$ is the mean-field at $v$ .

1.3. Markov chain mixing and cutoff phenomenon

The total variation distance between two probability measures $\nu_{1}$ and $\nu_{2}$ on $\Omega$ is defined by

\|\nu_{1}-\nu_{2}\|_{TV}\colonequals\sup_{A\subseteq\Omega}|\nu_{1}(A)-\nu_{2}(A)|=\frac{1}{2}\sum_{x\in\Omega}|\nu_{1}(x)-\nu_{2}(x)|.

The total variation distance is half of the $L^{1}$ -distance between the probability measures.

Let $({\sigma_{t}})$ be the Markov chain of the Glauber dynamics for the Ising model. Define the worst-case total variation distance of the chains to the stationary distribution $\mu$ at time $t$ by

d(t)\colonequals\max_{\sigma\in\Omega}\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\mu\|_{TV}

where here and thereafter $\mathbb{P}_{\sigma}$ denotes the probability given $\sigma_{0}=\sigma$ . The mixing time is defined by

t_{\mathrm{mix}}(\varepsilon)\colonequals\min\{t:d(t)\leq\varepsilon\};\quad\varepsilon\in(0,1).

We say a sequence of Markov chains with corresponding mixing times $t_{\mathrm{mix}}^{(n)}(\varepsilon)$ exhibit a cutoff phenomenon if for every $0<\varepsilon<1/2$ ,

\lim_{n\to\infty}\frac{t_{\mathrm{mix}}^{(n)}(\varepsilon)}{t_{\mathrm{m}ix}^{(n)}(1-\varepsilon)}=1.

Furthermore, we say that the cutoff occurs at $t_{\mathrm{mix}}^{(n)}$ with window size $O(w_{n})$ if $w_{n}=o(t_{\mathrm{mix}}^{(n)})$ and

\displaystyle\lim_{\gamma\to\infty}\liminf_{n\to\infty}d_{n}(t_{\mathrm{mix}}^{(n)}-\gamma w_{n})=1,\quad\lim_{\gamma\to\infty}\limsup_{n\to\infty}d_{n}(t_{\mathrm{mix}}^{(n)}+\gamma w_{n})=0.

1.4. Magnetization chain on complete multipartite graphs

Now, we are in a place to consider a complete $m$ -partite graph, a graph whose vertices are partitioned into $m$ different independent sets, and every pair of vertices from different independent sets is connected by an edge. Each edge represents an interaction between the vertices. Denote this graph by $K_{np_{1},np_{2},\dots,np_{m}}$ which has $n$ vertices and $m$ partitions where $\sum_{i=1}^{m}p_{i}=1$ and $p_{i}>0$ for $i=1,2,\dots,m$ . We fix the parameters $m$ and $p_{i}$ ’s hereafter. Without loss of generality, we assume $p_{1}\leq p_{2}\leq\dots\leq p_{m}$ . We may also assume that $np_{i}\in\mathbb{N}$ for every $i$ so that $K_{np_{1},np_{2},\dots,np_{m}}$ is well defined whenever such considerations are required. Let $V=\bigcup_{i=1}^{m}J_{i}$ be the set of all vertices where $J_{i}$ denotes the set of the $i$ th partition of the vertices. Note $np_{i}=|J_{i}|$ .

We define $\Omega_{i}\colonequals\{\pm 1\}^{J_{i}}$ for $i=1,\dots,m$ so that $\Omega=\prod_{i=1}^{m}\Omega_{i}$ is our configuration space. Each configuration $\sigma\in\Omega$ has a unique representation $(\sigma^{(1)},\dots,\sigma^{(m)})\in\prod_{i=1}^{m}\Omega_{i}$ and both representations are understood throughout this paper.

For each $\sigma\in\Omega$ , define the magnetization on $J_{i}$ by $S^{(i)}(\sigma)\colonequals\sum_{v\in J_{i}}\sigma(v)/n$ , $i=1,\dots,m$ . For the Markov chain $(\sigma_{t})_{t\geq 0}=(\sigma_{t}^{(1)},\dots,\sigma_{t}^{(m)})_{t\geq 0}$ starting at $\sigma=(\sigma^{(1)},\dots,\sigma^{(m)})\in\prod_{i=1}^{m}\Omega_{i}$ , we define the corresponding magnetization on $J_{i}$ by

S_{t}^{(i)}\colonequals\frac{1}{n}\sum_{v\in V_{i}}\sigma_{t}^{(i)}(v)\ \text{for}\ i\in\{1,\dots,m\},\ t\geq 0.

We sometimes use the vector notation $\mathbf{S}_{t}\colonequals(S^{(1)}_{t},\dots,S^{(m)}_{t})$ for $t\geq 0$ . We call the process $(\mathbf{S}_{t})_{t\geq 0}$ a magnetization chain. \threfmagmarkov shows that $(\mathbf{S}_{t})_{t\geq 0}$ is in fact a Markov chain. Note that it is a projection of the whole Markov chain $({\sigma_{t}})_{t\geq 0}$ , so mixing of the whole chain $({\sigma_{t}})_{t\geq 0}$ implies the mixing of the chain $(\mathbf{S}_{t})_{t\geq 0}$ . Our aim is to show the converse in a certain sense.

1.5. Main results

Given the above definitions and notations, our main result establishes the cutoff phenomenon on complete multipartite graphs.

Theorem 1.1 (Main result).

For $m\in\mathbb{N}$ and $p_{i}>0$ such that $\sum_{i=1}^{m}p_{i}=1$ , the Glauber dynamics for the Ising model on the complete multipartite graph $K_{np_{1},\dots,np_{m}}$ exhibits a cutoff at $\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ with window size $O(n)$ in the high temperature regime $\beta<\beta_{cr}$ where $\beta_{cr}=\beta_{cr}(p_{1},\dots,p_{m})$ is a constant defined in equation (3).

Theorem 1.2.

\thlabel

lowtempresult In the low temperature regime $\beta>\beta_{cr}$ , the dynamics is exponentially slow mixing, i.e., $t_{\mathrm{mix}}\geq C_{1}\exp(C_{2}n)$ for some constants $C_{1},\ C_{2}>0$ not depending on $n$ .

A few remarks are in order. Our main result is obtained as a consequence of \threfupperbound and \threflowerbound. In the low temperature regime $\beta>\beta_{cr}$ , the mixing time is exponentially slow, therefore identifying the critical temperature $\beta_{cr}$ . In the $m=1$ case, there are no spin interactions so the chain is equivalent to the lazy random walk on an $n$ -dimensional hypercube, which has a cutoff at $(n\ln n)/2$ with window size $O(n)$ (see [1] or [11, Chapter 18]). This result can be seen as a consequence of our main result since $m=1$ implies $\beta_{cr}=\infty$ (see equation (3)).

1.6. Organization of the article

As mentioned earlier, our proof is based on the ideas of [10]. We assume high temperatures until Section 6. We first observe that the magnetization chain is a Markov chain in its own right (\threfmagmarkov). A suitable scaling of the magnetization chain leads to a contraction property (\threfnormcontraction). This in turn gives a uniform variance bound of magnetizations in time (Sections 2 and 3). In Section 4, we construct a coupling of the magnetization chain so that it couples in $\frac{1}{2(1-\beta/\beta_{cr})}n\ln n+O(n)$ steps with high probability. After the magnetization coupling phase, by considering the " $2m$ -coordinate chain" inspired by [10], we can construct a post magnetization coupling to reach the full-mixing in another $O(n)$ steps. This proves the upper bound (\threfupperbound). We construct a suitable distinguishing-statistic of the magnetization chain [[, see]Chapter 7.3]Peres to obtain the lower bound (\threflowerbound). These upper and lower bound results establish the cutoff in the high temperature regime. Exponentially slow mixing in the low temperature regime is shown in Section 6.

2. Contraction of the magnetization chain in high temperatures

We describe the monotone coupling. Let $I$ and $U$ be independent uniform random variables over $V$ and $[0,1]$ , respectively. We consider the collection of Markov chains with starting configurations $\sigma\in\Omega$ . Simultaneously define the next configurations at time $t=1$ by

\sigma_{1}(i)=\begin{cases}\sigma(i)&\text{if $I\neq i$}\\ \mathbbm{1}_{U<r_{+}(\sum_{j\neq k}S^{(j)}(\sigma))}-\mathbbm{1}_{U\geq r_{+}(\sum_{j\neq k}S^{(j)}(\sigma))}&\text{if $I=i\in J_{k}$}\end{cases}\quad

where $r_{+}$ is defined in equation (1). Repeat this procedure independently for each time. It is clear that each Markov chain $(\sigma_{t})_{t\geq 0}$ above is a version of the Glauber dynamics on the complete multipartite graph with starting state $\sigma$ ’s, defined on a common probability space. The above coupling is called a monotone coupling in the sense that if $\sigma\leq\tilde{\sigma}$ are starting states for $(\sigma_{t})_{t\geq 0}$ and $(\tilde{\sigma}_{t})_{t\geq 0}$ , respectively, then $S^{(i)}(\sigma)\leq S^{(i)}(\tilde{\sigma})$ for $i=1,\dots,m$ so that $\sigma_{1}\leq\tilde{\sigma}_{1}$ , and $\sigma_{t}\leq\tilde{\sigma}_{t}$ for any $t\geq 0$ accordingly.

Define

\mathcal{S}\colonequals\prod_{i=1}^{m}\{-p_{i},-p_{i}+2/n,\dots,p_{i}\}.

Proposition 2.1 (Magnetization chain).

\thlabel

magmarkov The process $(S^{(1)}_{t},\dots,{S^{(m)}_{t}})_{t\geq 0}$ is a
Markov chain on the magnetization state space $\mathcal{S}$ .

Proof.

Note that

	$\displaystyle\mathbb{P}\bigl{(}({S^{(1)}_{t+1}},\dots,{S^{(m)}_{t+1}})=(S^{(1)}_{t}-\frac{2}{n},\dots,{S^{(m)}_{t}})\bigr{)}$	$\displaystyle=p_{1}\frac{n{S^{(1)}_{t}}+\|J_{1}\|}{2\|J_{1}\|}r_{-}\Bigl{(}\sum_{j\neq 1}{S^{(j)}_{t}}\Bigr{)}$
		$\displaystyle=\frac{p_{1}+{S^{(1)}_{t}}}{2}r_{-}\Bigl{(}\sum_{j\neq 1}{S^{(j)}_{t}}\Bigr{)}$

is measurable with respect to the $\sigma$ -algebra generated by $({S^{(1)}_{t}},\dots,{S^{(m)}_{t}})$ . Other cases can be dealt with similarly. ∎

Remark.

By symmetry, $({S^{(1)}_{t}},\dots,{S^{(m)}_{t}})$ starting from $\sigma$ and $(-{S^{(1)}_{t}},\dots,-{S^{(m)}_{t}})$ starting from $-\sigma$ have the same distributions. This can also be seen by the physical fact that the map $\sigma\mapsto-\sigma$ just corresponds to flipping the reference axis to which we are measuring the spins of each site. This does not change the dynamics of the spin system.

Definition (Hamming distance).

For two configurations $\sigma$ and $\sigma^{\prime}$ , denote the Hamming distance by $\mathrm{dist}(\sigma,\sigma^{\prime})\colonequals\frac{1}{2}\sum_{k\in V}|\sigma(k)-\sigma^{\prime}(k)|$ .

Remark.

This is a metric on $\Omega$ , which is equal to the number of sites with different spins for two configurations. Similarly, we can define $\mathrm{dist}_{i}$ on $\Omega_{i}$ , respectively, but $\mathrm{dist}_{i}$ ’s merely satisfy the triangle inequality.

Lemma 2.2 (Contraction in mean for monotone coupling).

\thlabel

monotonecontraction For a monotone coupling $({\sigma_{t}},{\sigma^{\prime}_{t}})_{t\geq 0}$ starting at $(\sigma,\sigma^{\prime})=((\sigma^{(1)},\dots,\sigma^{(2)}),(\sigma^{\prime(1)},\dots,\sigma^{\prime(2)}))$ , we have

\begin{pmatrix}\mathbb{E}\mathrm{dist}_{1}({\sigma^{(1)}_{t}},{\sigma^{\prime(1)}_{t}})\\ \vdots\\ \mathbb{E}\mathrm{dist}_{m}({\sigma^{(m)}_{t}},{\sigma^{\prime(m)}_{t}})\end{pmatrix}\leq\mathbf{A}^{t}\begin{pmatrix}\mathrm{dist}_{1}(\sigma^{(1)},\sigma_{0}^{\prime})\\ \vdots\\ \mathrm{dist}_{m}(\sigma^{(m)},\sigma^{\prime(m)})\end{pmatrix}

where

\mathbf{A}=\mathbf{A}_{n}\colonequals\begin{pmatrix}a&b_{1}&b_{1}&\dots&b_{1}\\ b_{2}&a&b_{2}&\dots&b_{2}\\ b_{3}&b_{3}&a&\dots&b_{3}\\ \vdots&\dots&&&\vdots\\ b_{m}&\dots&&\dots&a\end{pmatrix}

with $a\colonequals 1-1/n$ , $b_{k}\colonequals p_{k}{\beta}/{n}$ .

Proof.

Assume $d(\sigma,\sigma^{\prime})=1$ with $-1=\sigma(v)=-\sigma^{\prime}(v)$ for some vertex $v$ . Note $\sigma\leq\sigma^{\prime}$ . Since we are considering a monotone coupling, it holds that for each $i=1,\dots,m$ ,

\displaystyle\mathrm{dist}_{i}({\sigma^{(i)}_{1}},{\sigma^{\prime(i)}_{1}})=\mathbbm{1}_{v\in J_{i}}(1-\mathbbm{1}_{I=v})+\mathbbm{1}_{v\notin J_{i}}(\mathbbm{1}_{I\in J_{i}}\mathbbm{1}_{B_{i}})

where

B_{i}=\Biggl{\{}r_{+}\biggl{(}\sum_{l\neq i}S_{l}(\sigma)\biggr{)}\leq U<r_{+}\biggl{(}\sum_{l\neq i}S_{l}(\sigma^{\prime})\biggr{)}\Biggl{\}}.

Note that

	$\displaystyle\mathbb{P}(B_{i})$	$\displaystyle=\frac{1}{2}\Biggl{(}\tanh\biggl{(}\beta\sum_{l\neq i}S_{l}(\sigma^{\prime})\biggr{)}-\tanh\biggl{(}\beta\sum_{l\neq i}S_{l}(\sigma)\biggr{)}\Biggr{)}$
		$\displaystyle=\frac{1}{2}\Biggl{(}\tanh\biggl{(}\beta\biggl{(}\sum_{l\neq i}S_{l}(\sigma)+\frac{2}{n}\biggr{)}\biggr{)}-\tanh\biggl{(}\beta\sum_{l\neq i}S_{l}(\sigma)\biggr{)}\Biggr{)}\mathbbm{1}_{v\notin J_{i}}$
		$\displaystyle\leq\tanh\frac{\beta}{n}\mathbbm{1}_{v\notin J_{i}}.$

Since $I$ and $U$ are independent, for $i=1,\dots,m$ ,

\displaystyle\mathbb{E}\mathrm{dist}_{i}({\sigma^{(i)}_{1}},{\sigma^{\prime(i)}_{1}})\leq\mathbbm{1}_{v\in J_{i}}(1-\frac{1}{n})+\mathbbm{1}_{v\notin J_{i}}p_{i}\tanh\frac{\beta}{n}.

Suppose $\mathrm{dist}(\sigma,\sigma^{\prime})=k>1$ . There exists $\sigma^{0}\colonequals\sigma$ , $\sigma^{1}$ , $\dots$ , $\sigma^{k}\colonequals\sigma^{\prime}$ such that $\mathrm{dist}(\sigma^{i},\sigma^{i+1})=1$ . By the triangular inequality for $\mathrm{dist}_{i}$ and the fact $\tanh(\beta/n)\leq\beta/n$ ,

\displaystyle\mathbb{E}\mathrm{dist}_{i}({\sigma^{(i)}_{1}},{\sigma^{\prime(i)}_{1}})\leq(1-\frac{1}{n})\mathrm{dist}_{i}(\sigma^{(i)},\sigma^{\prime(i)})+p_{i}\frac{\beta}{n}\sum_{l\neq i}\mathrm{dist}_{l}(\sigma^{(l)},\sigma^{\prime(l)}).

Furthermore, by the Markov property,

\mathbb{E}[\mathrm{dist}_{i}({\sigma^{(i)}_{t+1}},\sigma^{\prime(i)}_{t+1})|{\sigma_{t}},{\sigma^{\prime}_{t}}]\leq(1-\frac{1}{n})\mathrm{dist}_{i}({\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}})+\frac{p_{i}\beta}{n}\sum_{l\neq i}\mathrm{dist}_{l}({\sigma^{(l)}_{t}},{\sigma^{\prime(l)}_{t}}).

By taking expectation and putting $x_{i,t}\colonequals\mathbb{E}\mathrm{dist}_{i}({\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}})$ , we have

\displaystyle\begin{pmatrix}x_{1,t}\\ \vdots\\ x_{m,t}\end{pmatrix}\leq\mathbf{A}\begin{pmatrix}x_{1,t-1}\\ \vdots\\ x_{m,t-1}\end{pmatrix}.

Iterating gives

\begin{pmatrix}x_{1,t}\\ \vdots\\ x_{m,t}\end{pmatrix}\leq\mathbf{A}^{t}\begin{pmatrix}\mathrm{dist}_{1}(\sigma^{(1)},\sigma^{\prime(1)})\\ \vdots\\ \mathrm{dist}_{m}(\sigma^{(m)},\sigma^{\prime(m)})\end{pmatrix}.

∎

From now on, $\mathbf{A}$ (which depends on the number of vertices $n$ ) always denotes the matrix defined in \threfmonotonecontraction. Note that $\mathbf{A}$ is a positive matrix, so by the Perron-Frobenius theorem, there exists the largest eigenvalue $g=g_{n}>0$ with the left eigenvector $\mathbf{a}^{T}\colonequals(a_{1},\dots,a_{m})>\mathbf{0}$ normalized in $l^{1}$ norm. Note that $g$ has algebraic multiplicity $1$ (see [16, Section 8.2] for a proof), so $\mathbf{a}^{T}$ is unique.

We fix the following notations

	$\displaystyle\upsilon\colonequals n(1-g)\text{ and }$		(2)
	$\displaystyle\beta_{cr}\colonequals\frac{1}{(m-1)\sum_{i=1}^{m}a_{i}p_{i}}$		(3)

where $g$ and $(a_{1},\dots,a_{m})$ are defined in the previous paragraph. Another characterization of $\beta_{cr}$ is given in \threfslowmixinglemma. Insuk Seo commented¹¹1personal communication that it can also be characterized as the threshold value of $\beta$ that makes $\mathbf{K}$ positive definite where $\mathbf{K}$ is defined through the equation $\mathbf{A}=\mathbf{I}-\frac{1}{n}\mathbf{K}$ , $\mathbf{I}$ being the $m$ -by- $m$ identity matrix. \threfupsilon connects the quantities $\upsilon$ and $\beta_{cr}$ .

Proposition 2.3.

\thlabel

upsilon The left eigenvector $\mathbf{a}^{T}$ only depends on $p_{1},\dots,p_{m}$ . Moreover, $\upsilon$ only depends on $p_{1},\dots,p_{m},$ and $\beta$ through the following equation:

\upsilon=1-\beta(m-1)\sum_{i=1}^{m}a_{i}p_{i}.

Therefore, $\beta_{cr}$ only depends on $p_{1},\dots,p_{m}$ , and we have $\upsilon=1-\beta/\beta_{cr}$ .

Proof.

Since $g$ satisfies

	$\displaystyle 0$	$\displaystyle=(n/\beta)^{m}\det(\mathbf{A}-gI)=\det(n\mathbf{A}/\beta-ngI/\beta)$
		$\displaystyle=\det\begin{pmatrix}\frac{\upsilon-1}{\beta}&p_{1}&p_{1}&\dots&p_{1}\\ p_{2}&\frac{\upsilon-1}{\beta}&p_{2}&\dots&p_{2}\\ p_{3}&p_{3}&\frac{\upsilon-1}{\beta}&\dots&p_{3}\\ \vdots&\dots&&&\vdots\\ p_{m}&\dots&&\dots&\frac{\upsilon-1}{\beta}\end{pmatrix},$

it holds that $(\upsilon-1)/\beta$ is a root of a polynomial with coefficients only depending on $p_{1},\dots,p_{m}$ . Since $\mathbf{a}$ is in the kernel of the transpose of the above matrix, it only depends on $p_{1},\dots,p_{m}$ .

Finally, $g=\|\mathbf{A}^{T}\mathbf{a}\|_{1}=1-1/n+\frac{\beta}{n}(m-1)\sum_{k}a_{i}p_{i}$ implies $\upsilon=1-\beta(m-1)\sum_{i}a_{i}p_{i}$ .

∎

We collect further properties of the matrix $\mathbf{A}$ and its left eigenvector $\mathbf{a}^{T}$ in the next two lemmas.

Lemma 2.4.

\thlabel

finalfinallemma We have

a_{1}\geq\dots\geq a_{m}\ \text{and}\ \sum_{i=1}^{m}a_{i}p_{i}\leq\frac{1}{m}.

The equality in the latter holds if and only if $p_{1}=\dots=p_{m}$ .

Proof.

Recall that we assumed $p_{1}\leq\dots\leq p_{m}$ .

We claim that $a_{1}\geq\dots\geq a_{m}$ . To that end, fix $i<j$ . From $\mathbf{a}^{T}\mathbf{A}=g\mathbf{a}^{T}$ , we have $(1-\frac{1}{n})a_{i}+\frac{\beta}{n}\sum_{k\neq i}a_{k}p_{k}-ga_{i}=0=(1-\frac{1}{n})a_{j}+\frac{\beta}{n}\sum_{k\neq j}a_{k}p_{k}-ga_{j}$ . Then $(1-\frac{1}{n}-g-\frac{\beta p_{i}}{n})a_{i}=(1-\frac{1}{n}-g-\frac{\beta p_{j}}{n})a_{j}$ , i.e., $(\beta p_{i}+1-\upsilon)a_{i}=(\beta p_{j}+1-\upsilon)a_{j}$ . Thus, $p_{i}\leq p_{j}$ implies $a_{i}\geq a_{j}$ , proving the claim.

By Chebyshev’s sum inequality, since $a_{i}\geq a_{j}$ and $p_{i}\leq p_{j}$ whenever $i<j$ ,

\displaystyle\sum_{i=1}^{m}a_{i}p_{i}\leq\frac{1}{m}\left(\sum_{i=1}^{m}a_{i}\right)\left(\sum_{i=1}^{m}p_{i}\right)=\frac{1}{m}.

The equality holds if and only if $a_{1}=\dots=a_{m}$ or $p_{1}=\dots=p_{m}$ . The proof is now complete by noticing the fact that $(\beta p_{i}+1-\upsilon)a_{i}=(\beta p_{j}+1-\upsilon)a_{j}$ and $a_{1}=\dots=a_{m}=1/m$ imply $p_{1}=\dots=p_{m}$ . ∎

Remark.

As a consequence, we obtain a lower bound $\beta_{cr}\geq m/(m-1)$ .

Lemma 2.5.

\thlabel

newlemma For $\mathbf{0}\leq\mathbf{s}\in\mathcal{S}$ and $\mathbf{p}\colonequals(p_{1},\dots,p_{m})^{T}$ , we have

\|\mathbf{A}^{t}\mathbf{s}\|_{1}\leq g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2},\quad\mathbf{e}_{j}^{T}\mathbf{A}^{t}\mathbf{s}\leq\sqrt{p_{j}}g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2}.

In particular, it holds that

\|\mathbf{A}^{t}\mathbf{p}\|_{1}\leq g^{t},\quad\mathbf{e}_{j}^{T}\mathbf{A}^{t}\mathbf{p}\leq\sqrt{p_{j}}g^{t}.

Proof.

We want to find a symmetric matrix $\mathbf{C}$ which is similar to $\mathbf{A}$ . To that end, suppose that there exists an invertible diagonal matrix $\mathbf{D}=\mathrm{diag}(d_{1},\dots,d_{m})$ and a symmetric matrix $\mathbf{C}$ such that $\mathbf{C}=\mathbf{D}^{-1}\mathbf{A}\mathbf{D}$ . Then $\mathbf{D}\mathbf{A}^{T}\mathbf{D}^{-1}=\mathbf{C}^{T}=\mathbf{C}=\mathbf{D}^{-1}\mathbf{A}\mathbf{D}$ , so $\mathbf{D}^{2}\mathbf{A}^{T}=\mathbf{A}\mathbf{D}^{2}$ , which leads to $d_{i}^{2}p_{j}=p_{i}d_{j}^{2}$ for $i,j\in\{1,2,\dots,m\}$ . With the above in mind, let $\mathbf{D}\colonequals\mathrm{diag}(\sqrt{p_{1}},\dots,\sqrt{p_{m}})$ and $\mathbf{C}\colonequals(c_{ij})$ where $c_{ii}=1-1/n$ and $c_{ij}=\sqrt{p_{i}p_{j}}{\beta}/{n}$ for $i\neq j$ . Note that $\mathbf{C}$ is real-symmetric and $\mathbf{C}=\mathbf{D}^{-1}\mathbf{A}\mathbf{D}$ . Then, by the spectral theorem for real symmetric matrices, $\|\mathbf{C}\|_{2}=g$ . Note that $\mathbf{C}$ and $\mathbf{A}$ have the same real eigenvalues since they are similar.

Observe that for $\mathbf{x},\mathbf{y}\in\mathbb{R}^{m}$ , $\|\mathbf{x}\mathbf{y}^{T}\|_{2}=\|\mathbf{x}\|_{2}\|\mathbf{y}\|_{2}$ . This can be easily checked by the equalities

\|\mathbf{x}\mathbf{y}^{T}\|_{2}=\sup_{\|\mathbf{z}\|_{2}=1}\|\mathbf{x}\mathbf{y}^{T}\mathbf{z}\|_{2}=\sup_{\|\mathbf{z}\|_{2}=1}|\mathbf{y}^{T}\mathbf{z}|\|\mathbf{x}\|_{2}=\|\mathbf{y}\|_{2}\|\mathbf{x}\|_{2}.

Let $\mathbbm{1}\colonequals(1,\dots,1)^{T}$ . The case $\mathbf{s}=\mathbf{0}$ is trivial, so assume $\mathbf{s}>\mathbf{0}$ . Since $\mathbf{s}\mathbbm{1}^{T}$ has rank 1, $\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}\mathbbm{1}^{T}\mathbf{D}$ has rank $1$ . Also, its elements are positive, so it has a positive eigenvalue by the Perron-Frobenius theorem. Thus, $\mathrm{Tr}(\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}\mathbbm{1}^{T}\mathbf{D})$ is equal to its spectral radius, from which the following inequality follows:

	$\displaystyle\\|\mathbf{A}^{t}\mathbf{s}\\|_{1}$	$\displaystyle=\mathbbm{1}^{T}\mathbf{A}^{t}\mathbf{s}=\mathbbm{1}^{T}\mathbf{D}\cdot\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}=\mathrm{Tr}(\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}\mathbbm{1}^{T}\mathbf{D})\leq\\|\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}\mathbbm{1}^{T}\mathbf{D}\\|_{2}$
		$\displaystyle\leq\\|\mathbf{C}\\|_{2}^{t}\\|\mathbf{D}^{-1}\mathbf{s}\mathbbm{1}^{T}\mathbf{D}\\|_{2}=g^{t}\\|\mathbf{D}^{-1}\mathbf{s}\\|_{2}\\|\mathbf{D}\mathbbm{1}\\|_{2}=g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2}.$

Similarly,

\mathbf{e}_{j}^{T}\mathbf{A}^{t}\mathbf{s}\leq\|\mathbf{C}\|_{2}^{t}\|\mathbf{D}^{-1}\mathbf{s}\mathbf{e}_{j}^{T}\mathbf{D}\|_{2}=g^{t}\|\mathbf{D}^{-1}\mathbf{s}\|_{2}\|\mathbf{D}\mathbf{e}_{j}\|_{2}=\sqrt{p_{j}}g^{t}\Biggl{(}\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\Biggr{)}^{1/2}.

∎

Remark.

Another relatively simple proof of $\|\mathbf{A}^{t}\mathbf{s}\|_{1}\leq g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2}$ can be given as follows. By the Cauchy-Schwartz inequality, we have $\sqrt{\sum_{i}s_{i}^{2}/p_{i}}\geq\sum_{i}s_{i}$ . Then $\|\mathbf{A}^{t}\mathbf{s}\|_{1}\leq\|\mathbf{D}^{-1}\mathbf{A}^{t}\mathbf{s}\|_{2}=\|\mathbf{D}^{-1}\mathbf{A}^{t}\mathbf{D}\mathbf{D}^{-1}\mathbf{s}\|_{2}=\|\mathbf{C}^{t}\mathbf{D}^{-1}\mathbf{s}\|_{2}\leq g^{t}\|\mathbf{D}^{-1}\mathbf{s}\|_{2}$ .

From now on, for brevity, we use the notation

\mathbf{p}\colonequals(p_{1},\dots,p_{m})^{T}.

Lemma 2.6.

\thlabel

lemma For a monotone coupling $({\sigma_{t}},{\sigma^{\prime}_{t}})_{t\geq 0}$ starting at $(\sigma,\sigma^{\prime})$ , we have

\mathbb{E}\sum_{i=1}^{m}a_{i}\mathrm{dist}_{i}(\sigma_{t},\sigma^{\prime}_{t})\leq g^{t}\sum_{i=1}^{m}a_{i}\mathrm{dist}_{i}(\sigma,\sigma^{\prime}).

Moreover, for $i=1,\dots,m$ ,

\mathbb{E}\mathrm{dist}_{i}({\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}})\leq n\sqrt{p_{i}}g^{t}.

Proof.

From \threfmonotonecontraction,

	$\displaystyle\mathbb{E}\sum_{i=1}^{m}a_{i}\mathrm{dist}_{i}(\sigma_{t},\sigma^{\prime}_{t})$	$\displaystyle=\mathbf{a}^{T}\begin{pmatrix}\mathbb{E}\mathrm{dist}_{1}({\sigma^{(1)}_{t}},{\sigma^{\prime(1)}_{t}})\\ \vdots\\ \mathbb{E}\mathrm{dist}_{m}({\sigma^{(m)}_{t}},{\sigma^{\prime(m)}_{t}})\end{pmatrix}\leq\mathbf{a}^{T}\mathbf{A}^{t}\begin{pmatrix}\mathrm{dist}_{1}(\sigma^{(1)},\sigma^{\prime(1)})\\ \vdots\\ \mathrm{dist}_{m}(\sigma^{(m)},\sigma^{\prime(m)})\end{pmatrix}$
		$\displaystyle\leq g^{t}\mathbf{a}^{T}\begin{pmatrix}\mathrm{dist}_{1}(\sigma^{(1)},\sigma^{\prime(1)})\\ \vdots\\ \mathrm{dist}_{m}(\sigma^{(m)},\sigma^{\prime(m)})\end{pmatrix}\leq g^{t}\sum_{i=1}^{m}a_{i}\mathrm{dist}_{i}(\sigma,\sigma^{\prime}).$

Notice that $\mathrm{dist}_{k}({\sigma^{(k)}_{t}},{\sigma^{\prime(k)}_{t}})\leq np_{k}$ for each $k$ , so \threfnewlemma implies

\mathbb{E}\mathrm{dist}_{i}({\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}})\leq n\mathbf{e}_{i}^{T}\mathbf{A}^{t}\mathbf{p}\leq n\sqrt{p_{i}}g^{t}.

∎

We would like to translate \threflemma to the case of magnetization chains, which is done in \threfnormcontraction.

Lemma 2.7.

\thlabel

generalcontraction For starting magnetizations $\mathbf{s}=(s^{(1)},\dots,s^{(m)})\geq(s^{\prime(1)},\dots,s^{\prime(m)})=\mathbf{s}^{\prime}$ , the magnetization chains satisfy

\mathbf{0}\leq\begin{pmatrix}\mathbb{E}_{\mathbf{s}}{S^{(1)}_{t}}-\mathbb{E}_{\mathbf{s}^{\prime}}{S^{\prime(1)}_{t}}\\ \vdots\\ \mathbb{E}_{\mathbf{s}}{S^{(m)}_{t}}-\mathbb{E}_{\mathbf{s}^{\prime}}{S^{\prime(m)}_{t}}\end{pmatrix}\leq\mathbf{A}^{t}\begin{pmatrix}s^{(1)}-s^{\prime(1)}\\ \vdots\\ s^{(m)}-s^{\prime(m)}\end{pmatrix}.

Remark.

We say such pairs of starting magnetizations are monotone pairs.

Proof.

Let $({\sigma_{t}},{\sigma^{\prime}_{t}})$ be a monotone coupling starting from $(\sigma,\sigma^{\prime})$ where $\sigma\geq\sigma^{\prime}$ and $S^{(i)}(\sigma)=s_{i}$ , $S^{\prime(i)}(\sigma^{\prime})=s_{i}^{\prime}$ for $i=1,\dots,m$ . Such a monotone coupling exists because of the given condition $s_{i}\geq s_{i}^{\prime}$ for each $i$ . Since $\sigma_{i}\geq\sigma_{i}^{\prime}$ , we have $s_{i}-s_{i}^{\prime}=\frac{2}{n}\mathrm{dist}_{i}(\sigma_{i},\sigma_{i}^{\prime})$ for each $i$ . By monotonicity, ${\sigma^{(i)}_{t}}\geq{\sigma^{\prime(i)}_{t}}$ for each $i$ . Thus, ${S^{(i)}_{t}}-{S^{\prime(i)}_{t}}=|{S^{(i)}_{t}}-{S^{\prime(i)}_{t}}|=\frac{2}{n}\mathrm{dist}_{i}({\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}})\geq 0$ for each $i$ . Then, by \threfmonotonecontraction,

\mathbf{0}\leq\begin{pmatrix}\mathbb{E}_{\sigma}{S^{(1)}_{t}}-\mathbb{E}_{\sigma^{\prime}}{S^{\prime(1)}_{t}}\\ \vdots\\ \mathbb{E}_{\sigma}{S^{(m)}_{t}}-\mathbb{E}_{\sigma^{\prime}}{S^{\prime(m)}_{t}}\end{pmatrix}=\begin{pmatrix}\mathbb{E}_{\sigma,\sigma^{\prime}}|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}|\\ \vdots\\ \mathbb{E}_{\sigma,\sigma^{\prime}}|{S^{(m)}_{t}}-{S^{\prime(m)}_{t}}|\end{pmatrix}\leq\mathbf{A}^{t}\begin{pmatrix}s^{(1)}-s^{\prime(1)}\\ \vdots\\ s^{(m)}-s^{\prime(m)}\end{pmatrix}.

Now, we can complete the proof since we have $\mathbb{E}_{\sigma}{S^{(i)}_{t}}-\mathbb{E}_{\sigma^{\prime}}{S^{\prime(i)}_{t}}=\mathbb{E}_{\mathbf{s}}{S^{(i)}_{t}}-\mathbb{E}_{\mathbf{s}^{\prime}}{S^{\prime(i)}_{t}}$ for each $i$ by \threfmagmarkov. ∎

Recall that $\circ$ denotes a Hadamard product.

Proposition 2.8.

\thlabel

normcontraction For a monotone coupling $(\sigma_{t},\sigma_{t}^{\prime})_{t\geq 0}$ starting at $(\sigma,\sigma^{\prime})$ with magnetizations $(\mathbf{s},\mathbf{s}^{\prime})$ , we have

\displaystyle\mathbb{E}_{\sigma,\sigma^{\prime}}\|\mathbf{a}\circ\mathbf{S}_{t}-\mathbf{a}\circ\mathbf{S}_{t}^{\prime}\|_{1}\leq g^{t}\|\mathbf{a}\circ\mathbf{s}-\mathbf{a}\circ\mathbf{s}^{\prime}\|_{1}.

Moreover, not depending on the coupling, we have

\displaystyle\|\mathbb{E}_{\mathbf{s}}\mathbf{a}\circ\mathbf{S}_{t}-\mathbb{E}_{\mathbf{s}^{\prime}}\mathbf{a}\circ\mathbf{S}_{t}^{\prime}\|_{1}\leq g^{t}\|\mathbf{a}\circ\mathbf{s}-\mathbf{a}\circ\mathbf{s}^{\prime}\|_{1}.

Proof.

For any magnetizations $\mathbf{s}\equiv\mathbf{s}_{(0)}$ and $\mathbf{s}^{\prime}\equiv\mathbf{s}_{(m)}$ , there exists $\mathbf{s}_{(1)},\dots,\mathbf{s}_{(m-1)}\in\mathcal{S}\subset\mathbb{R}^{m}$ such that $\mathbf{s}_{(i-1)}-\mathbf{s}_{(i)}=\mathbf{e}_{i}(s^{(i)}-s^{\prime(i)})$ for $i=1,\dots,m$ . In particular, $\mathbf{s}_{(i-1)}$ and $\mathbf{s}_{(i)}$ are a monotone pair for each $i$ . Then we can consider a monotone coupling $(\sigma_{(0),t},\dots,\sigma_{(m),t})_{t\geq 0}$ with starting states $(\sigma_{(0)},\dots,\sigma_{(m)})$ such that $\sigma_{t}=\sigma_{(0),t}$ , $\sigma_{t}^{\prime}=\sigma_{(m),t}$ for $t\geq 0$ , and the magnetization of the starting configuration $\sigma_{(i)}$ is $\mathbf{s}_{(i)}$ for $i=0,\dots,m$ .

Let $\mathbf{S}_{(j),t}$ be the magnetization chain corresponding to $\sigma_{(j),t}$ for $j=0,\dots,m$ . By telescoping, \threfgeneralcontraction gives

	$\displaystyle\mathbb{E}_{\sigma,\sigma^{\prime}}\\|\mathbf{a}\circ\mathbf{S}_{t}-\mathbf{a}\circ\mathbf{S}_{t}^{\prime}\\|_{1}\leq\sum_{j=1}^{m}\mathbb{E}_{\sigma_{(j-1)},\sigma_{(j)}}\\|\mathbf{a}\circ\mathbf{S}_{(j-1),t}-\mathbf{a}\circ\mathbf{S}_{(j),t}\\|_{1}$
	$\displaystyle\leq\sum_{j=1}^{m}\mathbf{a}^{T}\mathbf{A}^{t}\mathbf{e}_{j}\|s^{(j)}-s^{\prime(j)}\|=g^{t}\sum_{j=1}^{m}a_{j}\|s^{(j)}-s^{\prime(j)}\|=g^{t}\\|\mathbf{a}\circ\mathbf{s}-\mathbf{a}\circ\mathbf{s}^{\prime}\\|_{1}.$

Then, the triangle inequality and \threfmagmarkov imply

\|\mathbb{E}_{\mathbf{s}}\mathbf{a}\circ\mathbf{S}_{t}-\mathbb{E}_{\mathbf{s}^{\prime}}\mathbf{a}\circ\mathbf{S}_{t}^{\prime}\|_{1}\leq g^{t}\|\mathbf{a}\circ\mathbf{s}-\mathbf{a}\circ\mathbf{s}^{\prime}\|_{1}.

∎

3. Variance bound of the magnetization in high temperatures

The next lemma is a generalization of Lemma 2.6 in [10] to Markov chains with a finite state space in $\mathbb{R}^{m}$ . Observe that for square-integrable $\mathbb{R}^{m}$ -valued i.i.d. random vectors $X,Y$ , we have $\mathbb{V}\mathrm{ar}X=\frac{1}{2}\mathbb{E}\|X-Y\|_{2}^{2}$ .

Lemma 3.1.

\thlabel

variancebound Let $(\mathbf{Z}_{t})_{t\geq 0}$ be a Markov chain in a finite state space $\tilde{\mathcal{S}}\subseteq\mathbb{R}^{m}$ . Suppose that there exists $0<r<1$ such that for any $\theta,\theta^{\prime}\in\tilde{\mathcal{S}}$ ,

\|\mathbb{E}_{\theta}\mathbf{Z}_{t}-\mathbb{E}_{\theta^{\prime}}\mathbf{Z}_{t}^{\prime}\|_{1}\leq r^{t}\|\theta-\theta^{\prime}\|_{1}.

Then, for the $l^{2}$ norm variance,

\sup_{\theta\in\mathcal{S}}\mathbb{V}\mathrm{ar}_{\theta}\mathbf{Z}_{t}\leq m\sup_{\theta\in\mathcal{S}}\mathbb{V}\mathrm{ar}_{\theta}\mathbf{Z}_{1}\>\min\{t,(1-r^{2})^{-1}\}.

Proof.

Put $v_{t}\colonequals\sup_{\theta\in\mathcal{S}}{\mathbb{V}\mathrm{ar}}_{\theta}\mathbf{Z}_{t}$ . Let $(\mathbf{Z}_{t})$ and $(\mathbf{Z}_{t}^{\prime})$ be independent copies of the chain starting from $\theta\in\tilde{\mathcal{S}}$ . The idea is to condition on the first step. Note that $\|\mathbf{x}\|_{2}\leq\|\mathbf{x}\|_{1}\leq\sqrt{m}\|\mathbf{x}\|_{2}$ for $\mathbf{x}\in\mathbb{R}^{m}$ . Then by the observation right before the statement of this lemma,

\frac{1}{2}\mathbb{E}_{\theta}\|\mathbf{Z}_{1}-\mathbf{Z}_{1}^{\prime}\|_{1}^{2}\leq m\frac{1}{2}\mathbb{E}_{\theta}\|\mathbf{Z}_{1}-\mathbf{Z}_{1}^{\prime}\|_{2}^{2}\leq mv_{1}.

By the assumption and Markov property, we have

\|\mathbb{E}_{\theta}[\mathbf{Z}_{t}|\mathbf{Z}_{1}]-\mathbb{E}_{\theta}[\mathbf{Z}_{t}^{\prime}|\mathbf{Z}_{1}^{\prime}]\|_{1}=\|\mathbb{E}_{\mathbf{Z}_{1}}[\mathbf{Z}_{t-1}]-\mathbb{E}_{\mathbf{Z}_{1}^{\prime}}[\mathbf{Z}_{t-1}^{\prime}]\|_{1}\leq r^{t-1}\|\mathbf{Z}_{1}-\mathbf{Z}_{1}^{\prime}\|_{1}.

Thus, for $\theta\in\tilde{\mathcal{S}}$ ,

	$\displaystyle\mathbb{V}\mathrm{ar}_{\theta}[\mathbb{E}_{\theta}(\mathbf{Z}_{t}\|\mathbf{Z}_{1})]$	$\displaystyle=\frac{1}{2}\mathbb{E}_{\theta}\\|\mathbb{E}_{\mathbf{Z}_{1}}\mathbf{Z}_{t-1}-\mathbb{E}_{\mathbf{Z}_{1}^{\prime}}\mathbf{Z}_{t-1}^{\prime}\\|_{2}^{2}\leq\frac{1}{2}\mathbb{E}_{\theta}\\|\mathbb{E}_{\mathbf{Z}_{1}}\mathbf{Z}_{t-1}-\mathbb{E}_{\mathbf{Z}_{1}^{\prime}}\mathbf{Z}_{t-1}^{\prime}\\|_{1}^{2}$
		$\displaystyle\leq\frac{1}{2}\mathbb{E}_{\theta}\Big{[}r^{2(t-1)}\\|\mathbf{Z}_{1}-\mathbf{Z}_{1}^{\prime}\\|_{1}^{2}\Big{]}\leq mv_{1}r^{2(t-1)}.$

By the Markov property, for every $\theta\in\tilde{\mathcal{S}}$ , $\mathbb{V}\mathrm{ar}_{\theta}[\mathbf{Z}_{t}|\mathbf{Z}_{1}]\leq v_{t-1}$ , so

\sup_{\theta\in\mathcal{S}}\mathbb{E}_{\theta}[\mathbb{V}\mathrm{ar}_{\theta}[\mathbf{Z}_{t}|\mathbf{Z}_{1}]]\leq v_{t-1}.

The total variance formula holds since we are using the $l^{2}$ norm. Thus, taking supremum over $\theta\in\tilde{\mathcal{S}}$ in the total variance formula $\mathbb{V}\mathrm{ar}_{\theta}\mathbf{Z}_{t}=\mathbb{E}_{\theta}\big{[}\mathbb{V}\mathrm{ar}_{\theta}[\mathbf{Z}_{t}|\mathbf{Z}_{1}]\big{]}+\mathbb{V}\mathrm{ar}_{\theta}\big{[}\mathbb{E}_{\theta}[\mathbf{Z}_{t}|\mathbf{Z}_{1}]\big{]}$ , we have $v_{t}\leq v_{t-1}+mv_{1}r^{2(t-1)}$ . Upon iterating,

v_{t}\leq mv_{1}\sum_{t=1}^{t}r^{2(t-1)}\leq mv_{1}\min\big{\{}t,(1-r^{2})^{-1}\big{\}}.

∎

The following proposition is an important result bounding the variance of magnetization chains uniformly in time.

Proposition 3.2.

\thlabel

magvariancebound Let $\beta<\beta_{cr}$ . For an arbitrary starting configuration $\mathbf{s}$ and $t\geq 0$ , we have

\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\mathbf{s}}({S^{(i)}_{t}})=C/n

where $C>0$ only depends on $p_{1},\dots,p_{m}$ , and $\beta$ .

Proof.

Observe that $\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\mathbf{s}}(a_{i}S^{(i)}_{t})=\mathbb{V}\mathrm{ar}_{\mathbf{s}}(\mathbf{a}\circ\mathbf{S}_{t})$ . Note that increments of $\mathbf{S}_{t}$ are bounded by $2/n$ in absolute value. Then, from \threffinalfinallemma, we have

\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\mathbf{s}}{a_{i}S^{(i)}_{1}}\leq a_{1}^{2}(2/n)^{2}.

By \threffinalfinallemma, \threfnormcontraction, and \threfvariancebound, we have

a_{m}^{2}\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\mathbf{s}}({S^{(i)}_{t}})\leq\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\mathbf{s}}(a_{i}{S^{(i)}_{t}})\leq m\frac{4a_{1}^{2}}{n^{2}}\frac{1}{1-g^{2}}=\frac{4ma_{1}^{2}}{\upsilon n(1+g)}\leq\frac{4ma_{1}^{2}}{\upsilon n}.

Note that \threfupsilon assures $\upsilon>0$ . ∎

We also establish a bound for the expected magnetization on subsets of partitions. To that end, we need the following observation.

Lemma 3.3.

\thlabel

zerospin For each $i\in V$ , $\mathbb{E}_{\mu}(\sigma(i))=0$ where $\mu$ is the Gibbs distribution. In particular, we have $\mathbb{E}_{\mu}(S^{(i)})=0$ .

Proof.

Since $\mu(\sigma)=\mu(-\sigma)$ for each configuration $\sigma$ and $\sigma\mapsto-\sigma$ is a bijection from $\Omega$ into itself, we have $\mathbb{E}_{\mu}(\sigma(i))=\sum_{\sigma}\sigma(i)\mu(\sigma)=\sum_{\sigma:\sigma(i)=1}\mu(\sigma)-\sum_{\sigma:\sigma(i)=-1}\mu(\sigma)=0$ . ∎

Proposition 3.4 (Expected magnetization bound).

\thlabel

expectedmagbound Let $\beta<\beta_{cr}$ and $1\leq i\leq m$ . For any $B\subseteq J_{i}$ and a chain $(\sigma_{t})_{t\geq 0}$ starting at $\sigma\in\Omega$ , define $M_{t}(B)\colonequals\frac{1}{2}\sum_{k\in B}{\sigma_{t}}(k)$ . Then

|\mathbb{E}_{\sigma}M_{t}(B)|\leq|B|g^{t}/\sqrt{p_{i}}.

Furthermore, for $t\geq\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ , we have

\mathbb{V}\mathrm{ar}_{\sigma}(M_{t}(B))=O(n)\ ,\quad\mathbb{E}_{\sigma}|M_{t}(B)|=O(\sqrt{n}).

Proof.

Let "+" denote the configuration such that all spins are $1$ and "-" denote the configuration with all spins $-1$ . Let $(\sigma_{t}^{+},\sigma_{t}^{\mu},\sigma_{t}^{-})$ be a monotone coupling with starting configuration $(+,\mu,-)$ where $\mu$ is the stationary distribution. Let $i\in\{1,\dots,m\}$ . By \threflemma and \threfzerospin,

\displaystyle\mathbb{E}_{+}[M_{t}(J_{i})^{+}]\leq\mathbb{E}_{+,\mu}|M_{t}(J_{i})^{+}-M_{t}(J_{i})^{\mu}|+\mathbb{E}_{\mu}[M_{t}(J_{i})^{\mu}]\leq n\sqrt{p_{i}}g^{t}.

Then, by symmetry, for $v\in J_{i}$ , $\mathbb{E}_{+}[M_{t}(v)]\leq n\sqrt{p_{i}}g^{t}/|J_{i}|=g^{t}/\sqrt{p_{i}}$ . Thus, by summing over sites in $B$ , $\mathbb{E}_{+}[M_{t}(B)^{+}]\leq|B|g^{t}/\sqrt{p_{i}}$ . However, for any configuration $\sigma$ , by monotonicity, $\mathbb{E}_{+}[M_{t}(B)^{+}]\geq\mathbb{E}_{\sigma}[M_{t}(B)]\geq\mathbb{E}_{-}[M_{t}(B)^{-}]$ . Considering the remark after \threfmagmarkov, $\mathbb{E}_{-}[M_{t}(B)^{-}]=-\mathbb{E}_{+}[M_{t}(B)^{+}]$ . Thus, $|\mathbb{E}_{\sigma}[M_{t}(B)]|\leq|\mathbb{E}_{+}[M_{t}(B)^{+}]|\leq|B|g^{t}/\sqrt{p_{i}}$ for any $\sigma$ .

Now, by \threfmagvariancebound, $O(1/n)=\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}=\mathbb{V}\mathrm{ar}(M_{t}(J_{i})2/n)$ , so

\mathbb{V}\mathrm{ar}_{+}(M_{t}(J_{i}))=O(n).

Thus, for $t\geq\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ ,

\mathbb{E}_{+}(M_{t}(J_{i})^{2})=\mathbb{V}\mathrm{ar}_{+}(M_{t}(J_{i}))+(\mathbb{E}_{+}M_{t}(J_{i}))^{2}=O(n)

However, by symmetry, for any fixed $v_{1},v_{2}\in J_{i}$ ,

\mathbb{E}_{+}(M_{t}(J_{i})^{2})=np_{i}+\binom{np_{i}}{2}\mathbb{E}_{+}(\sigma_{t}^{+}(v_{1})\sigma_{t}^{+}(v_{2})).

Thus,

|\mathbb{E}_{+}\sigma_{t}^{+}(v_{1})\sigma_{t}^{+}(v_{2})|=O(1/n).

Likewise, for $B\subseteq J_{i}$ ,

\mathbb{E}_{+}(M_{t}(B)^{2})=|B|+\binom{|B|}{2}\mathbb{E}_{+}(\sigma_{t}^{+}(v_{1})\sigma_{t}^{+}(v_{2}))\leq O(n).

Similarly, $\mathbb{E}_{-}M_{t}(B)^{2}\leq O(n)$ , so from $(M_{t}(B))^{2}\leq(M_{t}(B)^{+})^{2}+(M_{t}(B)^{-})^{2}$ ,

\mathbb{E}(M_{t}(B)^{2})=O(n)

whenever $t\geq\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ . Thus, for $t\geq\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ ,

\mathbb{V}\mathrm{ar}_{\sigma}(M_{t}(B))=O(n).

Lastly, for $t\geq\frac{1}{2(1-\beta/\beta_{cr})}n\ln n$ , from Jensen’s inequality,

	$\displaystyle\mathbb{E}_{\sigma}\|M_{t}(B)\|$	$\displaystyle\leq\sqrt{\mathbb{E}_{\sigma}\|M_{t}(B)\|^{2}}=\sqrt{(\mathbb{E}_{\sigma}[M_{t}(B)])^{2}+\mathbb{V}\mathrm{ar}_{\sigma}(M_{t}(B))}$
		$\displaystyle\leq\|\mathbb{E}_{\sigma}[M_{t}(B)]\|+\sqrt{\mathbb{V}\mathrm{ar}_{\sigma}(M_{t}(B))}=O(\sqrt{n}).$

∎

4. Couplings

Fix the notation

t_{n}\colonequals\frac{1}{2(1-\beta/\beta_{cr})}n\ln n.

Definition (Modified matching).

Let $\sigma\in\Omega$ and $\sigma^{\prime}\in\Omega$ have magnetizations $\mathbf{s}\in\mathcal{S}$ and $\mathbf{s}^{\prime}\in\mathcal{S}$ , respectively. Consider two copies of the graph, $V=\bigcup_{i}J_{i}$ and $V^{\prime}=\bigcup_{i}J_{i}^{\prime}$ . Let $i\in\{1,\dots,m\}$ . If $s^{(i)}\geq s^{\prime(i)}$ , then it is possible to match each site in $J_{i}^{\prime}$ with $+1$ spin to a site in $J_{i}$ with $+1$ spin. Any leftover sites in $J_{i}^{\prime}$ are arbitrarily matched to the leftover sites in $J_{i}$ . We match the sites in a similar way whenever $s^{(i)}\leq s^{\prime(i)}$ . This defines a bijection $f_{\sigma,\sigma^{\prime}}\colon V\to V^{\prime}$ .

We call this bijection a modified matching of $\sigma$ and $\sigma^{\prime}$ .

Definition (Modified monotone update and coupling).

Let $f_{\sigma,\sigma^{\prime}}\colon V\to V^{\prime}$ be a modified matching of $\sigma,\sigma^{\prime}\in\Omega$ . Let $I$ and $U$ be uniformly distributed over $V=\bigcup_{i=1}^{m}J_{i}$ and $[0,1]\subseteq\mathbb{R}$ , respectively, and be independent. Suppose $I\in J_{\eta}$ for some $\eta\in\{1,\dots,m\}$ is the chosen site in $V$ . Consider the case $\sum_{v\notin J_{\eta}}\sigma(v)\leq\sum_{v\notin J_{\eta}}\sigma^{\prime}(v)$ . If

U<\frac{1+\tanh\left(\beta\sum_{v\notin J_{\eta}}\sigma(v)\right)}{2},

then update the chosen site $I$ of $V$ by +1 and $f_{\sigma,\sigma^{\prime}}(I)$ of $V^{\prime}$ by +1. If

U\geq\frac{1+\tanh\left(\beta\sum_{v\notin J_{\eta}}\sigma^{\prime}(v)\right)}{2},

then update the chosen site $I$ of $V$ by -1 and $f_{\sigma,\sigma^{\prime}}(I)$ of $V^{\prime}$ by -1. Otherwise, if

\frac{1+\tanh\left(\beta\sum_{v\notin J_{\eta}}\sigma(v)\right)}{2}\leq U<\frac{1+\tanh\left(\beta\sum_{v\notin J_{\eta}}\sigma^{\prime}(v)\right)}{2},

then update the chosen site $I$ of $V$ by -1 and $f_{\sigma,\sigma^{\prime}}(I)$ of $V^{\prime}$ by +1. The other case $\sum_{v\notin J_{\eta}}\sigma(v)>\sum_{v\notin J_{\eta}}\sigma^{\prime}(v)$ can similarly be updated.

Given the chosen site $I$ , we call the above procedure of deciding the updating spin in the two chains a modified monotone update with respect to the given modified matching.

Now, fix a modified matching $f_{\sigma,\sigma^{\prime}}$ of $\sigma$ and $\sigma^{\prime}$ . Let ${\sigma_{t}}$ and ${\sigma^{\prime}_{t}}$ be chains starting at $\sigma$ and $\sigma^{\prime}$ , respectively. Repeating the above procedure independently for each step with respect to $f_{\sigma,\sigma^{\prime}}$ gives a coupling of the Glauber dynamics. We call this coupling a modified monotone coupling with respect to the given modified matching.

Remark.

\thref

monotonecontraction and its consequences hold with a suitable distance function for a modified coupling with respect to a given modified matching.

We first construct a coupling such that the magnetizations agree after $t_{n}+O(n)$ steps in the next two lemmas.

Lemma 4.1 (Lemma 2.4, [10]).

\thlabel

supermartingale Let $(W_{t})_{t\geq 0}$ be a non-negative supermartingale with a stopping time $\tau$ satisfying
(i) $W_{0}=k$
(ii) $W_{t+1}-W_{t}\leq B<\infty$
(iii) $\mathbb{V}\mathrm{ar}(W_{t+1}|\mathcal{F}_{t})>\sigma^{2}>0\ \text{on the event}\ \{\tau>t\}$ . Then for $u>\frac{4B^{2}}{3\sigma^{2}}$ ,

\mathbb{P}_{k}(\tau>u)\leq\frac{4k}{\sigma\sqrt{u}}.

Lemma 4.2 (Magnetization coupling).

\thlabel

magcoupling Let $\beta<\beta_{cr}$ . For any configurations $\sigma$ and $\sigma^{\prime}$ , there exists a coupling $({\sigma_{t}},{\sigma^{\prime}_{t}})$ with starting states $(\sigma,\sigma^{\prime})$ satisfying the following condition. If $\tau_{mag}\colonequals\min\{t\geq 0:\mathbf{S}_{t}=\mathbf{S}_{t}^{\prime}\}$ , then for large $\gamma n$ ,

\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau_{mag}>t_{n}+\gamma n)\leq\frac{c}{\sqrt{\gamma}}

where $c>0$ is a constant not depending on $\sigma$ , $\sigma^{\prime}$ , or $n$ .

Proof.

Let $({\sigma_{t}},{\sigma^{\prime}_{t}})$ be a monotone coupling with starting states $(\sigma,\sigma^{\prime})$ . Put $Y_{i,t}\colonequals\frac{n}{2}a_{i}|{S^{(i)}_{t}}-{S^{\prime(i)}_{t}}|$ for $i=1,\dots,m$ and $Y_{tot,t}\colonequals\sum_{i=1}^{m}Y_{i,t}$ . Define

\tau\colonequals\min\{t\geq t_{n}:\max_{1\leq i\leq m}Y_{i,t}/a_{i}\leq 1\}.

By \threfnormcontraction,

\mathbb{E}_{\sigma,\sigma^{\prime}}[Y_{tot,t_{n}}]\leq c\sqrt{n}

for some $c>0$ .

We construct a coupling such that $(Y_{tot,t})_{t_{n}\leq t<\tau}$ is a positive supermartingale with bounded increments and the conditional probability of not being lazy is bounded away from zero uniformly in time and $n$ .

To that end, consider a time $t_{n}\leq t<\tau$ . Define $K_{t}\colonequals\bigcup_{i:Y_{i,t}/a_{i}\leq 1}J_{i}$ , $L_{t}\colonequals\bigcup_{i:Y_{i,t}/a_{i}>1}J_{i}$ , and $L_{t}^{\prime}\colonequals\bigcup_{i:Y_{i,t}/a_{i}>1}J_{i}^{\prime}$ . Note that $L_{t}\neq\emptyset$ since $t<\tau$ . Choose a site equiprobably over $V=K_{t}\dot{\cup}L_{t}$ . Let $f_{t}$ be the modified matching of ${\sigma_{t}}$ and ${\sigma^{\prime}_{t}}$ . If a site in $K_{t}$ is chosen, then use the modified monotone update with respect to $f_{t}$ to update $({\sigma_{t}},{\sigma^{\prime}_{t}})$ . If a site in $L_{t}$ is chosen, then independently choose another site equiprobably over $L_{t}^{\prime}$ (which can be the same site) to update ${\sigma^{\prime}_{t}}$ independent of ${\sigma_{t}}$ . It is easy to check that the above is a coupling of the Glauber dynamics.

Clearly, $Y_{tot,t}$ has bounded increment with the above coupling. Let $I$ be a random variable uniformly distributed over $V$ which is independent of $\mathcal{F}_{t}$ . Let $E=\{I\in L_{t},{\sigma_{t}}(I)=+1,\sigma_{t+1}(I)=-1,\sigma_{t+1}^{\prime}(f_{t}(I))=1\}$ and $F=\{I\in L_{t},{\sigma_{t}}(I)=-1,\sigma_{t+1}(I)=+1,\sigma_{t+1}^{\prime}(f_{t}(I))=-1\}$ . Since $L_{t}\neq\emptyset$ implies $|L_{t}|/n\geq p_{1}$ , we obtain that $\mathbb{P}(Y_{tot,t+1}\neq Y_{tot,t}|\mathcal{F}_{t})$ is bounded below by

	$\displaystyle\geq\mathbb{P}(Y_{tot,t+1}\neq Y_{tot,t},I\in L_{t}\|\mathcal{F}_{t})\geq\mathbb{P}(E\dot{\cup}F\|\mathcal{F}_{t})$
	$\displaystyle\geq\frac{\|L_{t}\|+\sum_{i\in L_{t}}{\sigma_{t}}(i)}{2n}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}$
	$\displaystyle\enspace+\frac{\|L_{t}\|-\sum_{i\in L_{t}}{\sigma_{t}}(i)}{2n}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}$
	$\displaystyle\geq p_{1}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}>0.$

Finally, we need to show the supermartingale property. Consider $Y_{1,t+1}/a_{1}-Y_{1,t}/a_{1}$ . Suppose $J_{1}\subseteq K_{t}$ . Then by a direct calculation, on the event $\{J_{1}\subseteq K_{t}\}$ , it holds that $\mathbb{E}(Y_{1,t+1}/a_{1}-Y_{1,t}/a_{1}|\mathcal{F}_{t})$ is bounded above by

	$\displaystyle\leq\biggl{(}p_{1}-\frac{\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|}{2}\biggr{)}\frac{\|\tanh(\beta\sum_{j\neq 1}{S^{(j)}_{t}})-\tanh(\beta\sum_{j\neq 1}{S^{\prime(j)}_{t}})\|}{2}$
	$\displaystyle\enspace-\frac{\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|}{2}\biggl{(}1-\frac{\|\tanh(\beta\sum_{j\neq 1}{S^{(j)}_{t}})-\tanh(\beta\sum_{j\neq 1}{S^{\prime(j)}_{t}})\|}{2}\biggr{)}$
	$\displaystyle\leq\frac{1}{2}\biggl{(}-\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|+p_{1}\tanh\biggl{(}\beta\Bigl{\|}\sum_{j\neq 1}{S^{(j)}_{t}}-\sum_{j\neq 1}{S^{\prime(j)}_{t}}\Bigl{\|}\biggr{)}\biggr{)}.$

Suppose $J_{1}\subseteq L_{t}$ . Note that $Y_{1,t}>1$ implies $({S^{(1)}_{t+1}}-{S^{\prime(1)}_{t+1}})({S^{(1)}_{t}}-{S^{\prime(1)}_{t}})\geq 0$ and $|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}|>0$ . Let $\xi=({S^{(1)}_{t}}-{S^{\prime(1)}_{t}})/|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}|\in\{\pm 1\}$ . Then by equation (5) in Section 5.2, on the event $\{J_{1}\subseteq L_{t}\}$ , $\mathbb{E}(Y_{1,t+1}/a_{1}-Y_{1,t}/a_{1}|\mathcal{F}_{t})$ is equal to

	$\displaystyle=\xi\frac{n}{2}\biggl{(}\mathbb{E}({S^{(1)}_{t+1}}-{S^{(1)}_{t}}\|{\sigma_{t}})-\mathbb{E}({S^{\prime(1)}_{t+1}}-{S^{\prime(1)}_{t}}\|{\sigma^{\prime}_{t}})\biggr{)}$
	$\displaystyle=\xi\frac{n}{2}\frac{1}{n}\biggl{(}-{S^{(1)}_{t}}+p_{1}\tanh(\beta\sum_{j\neq 1}{S^{(j)}_{t}})\biggr{)}$
	$\displaystyle\quad-\xi\frac{n}{2}\frac{1}{n}\biggl{(}-{S^{\prime(1)}_{t}}+p_{1}\tanh(\beta\sum_{j\neq 1}{S^{\prime(j)}_{t}})\biggr{)}$
	$\displaystyle=\frac{\xi}{2}\biggl{(}-({S^{(1)}_{t}}-{S^{\prime(1)}_{t}})+p_{1}\biggl{(}\tanh(\beta\sum_{j\neq 1}{S^{(j)}_{t}})-\tanh(\beta\sum_{j\neq 1}{S^{\prime(j)}_{t}})\biggr{)}\biggr{)}$
	$\displaystyle\leq\frac{1}{2}\biggl{(}-\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|+p_{1}\tanh\biggl{(}\beta\Bigl{\|}\sum_{j\neq 1}{S^{(j)}_{t}}-\sum_{j\neq 1}{S^{\prime(j)}_{t}}\Bigl{\|}\biggr{)}\biggr{)}.$

Since either $J_{1}\subseteq L_{t}$ or $J_{1}\subseteq K_{t}$ must hold, $\mathbb{E}(Y_{1,t+1}/a_{1}-Y_{1,t}/a_{1}|\mathcal{F}_{t})$ is equal to

	$\displaystyle=\mathbbm{1}_{J_{1}\subseteq K_{t}}\mathbb{E}(Y_{1,t+1}-Y_{1,t}\|\mathcal{F}_{t})+\mathbbm{1}_{J_{1}\subseteq L_{t}}\mathbb{E}(Y_{1,t+1}-Y_{1,t}\|\mathcal{F}_{t})$
	$\displaystyle\leq\frac{1}{2}\biggl{(}-\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|+p_{1}\tanh\biggl{(}\beta\Bigl{\|}\sum_{j\neq 1}{S^{(j)}_{t}}-\sum_{j\neq 1}{S^{\prime(j)}_{t}}\Bigr{\|}\biggr{)}\biggr{)}$
	$\displaystyle\leq\frac{1}{2}\biggl{(}-\|{S^{(1)}_{t}}-{S^{\prime(1)}_{t}}\|+p_{1}\beta\sum_{j\neq 1}\Bigl{\|}{S^{(j)}_{t}}-{S^{\prime(j)}_{t}}\Bigl{\|}\biggr{)}.$

Thus,

\mathbb{E}(Y_{1,t+1}/a_{1}|\mathcal{F}_{t})\leq(1-\frac{1}{n})Y_{1,t}/a_{1}+\frac{\beta p_{1}}{n}\sum_{j\neq 1}Y_{j,t}/a_{j}.

Putting in the matrix form with $\tilde{\mathbf{Y}}_{t}\colonequals(Y_{1,t}/a_{1},\dots,Y_{m,t}/a_{m})^{T}$ , we have

\displaystyle\mathbb{E}(Y_{tot,t+1}|\mathcal{F}_{t})=\mathbf{a}^{T}\mathbb{E}(\tilde{\mathbf{Y}}_{t+1}|\mathcal{F}_{t})\leq\mathbf{a}^{T}\mathbf{A}\tilde{\mathbf{Y}}_{t}=g\mathbf{a}^{T}\tilde{\mathbf{Y}}_{t}=gY_{tot,t}.

Since $\beta<\beta_{cr}$ implies $g<1$ by \threfupsilon, the supermartingale property is established.

With the above coupling, by \threfsupermartingale, for large $\gamma n$ ,

\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau>t_{n}+\gamma n|\sigma_{t_{n}},\sigma^{\prime}_{t_{n}})\leq c^{\prime}\frac{n\|(S^{(1)}_{t_{n}},\dots,S^{(m)}_{t_{n}})-(S^{\prime(1)}_{t_{n}},\dots,S^{\prime(m)}_{t_{n}})\|_{1}}{\sqrt{\gamma n}}

for some $c^{\prime}>0$ not depending on $n$ . Taking expectation,

\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau>t_{n}+\gamma n)\leq O(\gamma^{-1/2}).

Note $\sigma_{\tau}$ has at most $m$ more +1 spin sites than $\sigma_{\tau}^{\prime}$ , so $0\leq Y_{tot,\tau}\leq a_{1}m$ by \threffinalfinallemma. At $\tau$ , construct a modified matching of $\sigma_{\tau}$ and $\sigma_{\tau}^{\prime}$ , and use the modified monotone coupling with respect to this modified matching from then on. At $\tau_{mag}$ , we construct another modified matching of the sites to do a new modified monotone coupling so that $({S^{(1)}_{t}},\dots,{S^{(m)}_{t}})=({S^{\prime(1)}_{t}},\dots,{S^{\prime(m)}_{t}})$ forever after $\tau_{mag}$ .

By \threffinalfinallemma, a modified version of \threfnormcontraction, and the strong Markov property, we have

	$\displaystyle\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau_{mag}>\tau+\gamma^{\prime}n\|\sigma_{\tau},\sigma_{\tau}^{\prime})$	$\displaystyle\leq\mathbb{P}_{\sigma,\sigma^{\prime}}(Y_{tot,\tau+\gamma^{\prime}n}\geq a_{m}\|\sigma_{\tau},\sigma_{\tau}^{\prime})$
		$\displaystyle\leq\mathbb{E}_{\sigma,\sigma^{\prime}}[Y_{tot,\tau+\gamma^{\prime}n}\|\sigma_{\tau},\sigma_{\tau}^{\prime}]/a_{m}$
		$\displaystyle\leq g^{\gamma^{\prime}n}Y_{tot,\tau}/a_{m}\leq g^{\gamma^{\prime}n}a_{1}m/a_{m}\leq e^{-\upsilon\gamma^{\prime}}a_{1}m/a_{m}.$

Thus,

\displaystyle\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau_{mag}>t_{n}+(\gamma+\gamma^{\prime})n)

\displaystyle\leq O(\gamma^{-1/2})+e^{-\upsilon\gamma^{\prime}}a_{1}m/a_{m},

and putting $\gamma=\gamma^{\prime}$ yields

\displaystyle\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau_{mag}>t_{n}+\gamma n)\leq O(\gamma^{-1/2}).

∎

Definition (Good configurations).

Define the set of "good" configurations by

\tilde{\Omega}\colonequals\{\sigma\in\Omega:|S^{(i)}(\sigma)|\leq p_{i}/2,\ i=1,\dots,m\}.

For $\sigma=(\sigma^{(1)},\dots,\sigma^{(m)})\in\tilde{\Omega}$ and each $i$ , define

\displaystyle u_{i}^{\sigma}\colonequals|\{v\in J_{i}:\sigma^{(i)}(v)=1\}|,\enspace v_{i}^{\sigma}\colonequals|\{v\in J_{i}:\sigma^{(i)}(v)=-1\}|.

Define

\tilde{\Lambda}\colonequals\{(u_{1},v_{1},u_{2},v_{2},\dots,u_{m},v_{m})\in\mathbb{N}^{2m}:{|J_{i}|}/{4}\leq u_{i}\wedge v_{i},\ i=1,\dots,m\}.

Remark.

Note that $\sigma\in\tilde{\Omega}\iff(u_{1}^{\sigma},v_{1}^{\sigma},\dots,u_{m}^{\sigma},v_{m}^{\sigma})\in\tilde{\Lambda}$ . In other words, $\tilde{\Lambda}$ is another representation of good configurations $\tilde{\Omega}$ . We omit the starting state and write $u_{i}$ instead of $u_{i}^{\sigma}$ for convenience.

Lemma 4.3 (Lemma 3.3, [10]).

\thlabel

goodstartingstatesdistance For any subset $A\subseteq\Omega$ and stationary distribution $\pi$ ,

	$\displaystyle d_{n}(t_{0}+t)$	$\displaystyle=\max_{\sigma\in\Omega}\\|\mathbb{P}_{\sigma}(\sigma_{t_{0}+t}\in\cdot)-\pi\\|_{TV}$
		$\displaystyle\leq\max_{\sigma\in A}\\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\pi\\|_{TV}+\max_{\sigma\in\Omega}\mathbb{P}_{\sigma}(\sigma_{t_{0}}\notin A).$

Recall that we are assuming the high temperature regime. By \threfexpectedmagbound, there exists $\delta>0$ such that $\max_{\sigma\in\Omega,1\leq i\leq m}|\mathbb{E}_{\sigma}S^{(i)}_{\delta n}|\leq p_{1}/4$ . Hence, by \threfmagvariancebound, for large $n$ ,

	$\displaystyle\mathbb{P}_{\sigma}(\sigma_{\delta n}\notin\tilde{\Omega})$	$\displaystyle\leq\sum_{i=1}^{m}\mathbb{P}_{\sigma}(\|S^{(i)}_{\delta n}\|>p_{i}/2)\leq\sum_{i=1}^{m}\mathbb{P}_{\sigma}(\|S^{(i)}_{\delta n}-\mathbb{E}_{\sigma}S^{(i)}_{\delta n}\|>p_{i}/4)$
		$\displaystyle\leq\frac{16}{p_{1}^{2}}\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}_{\sigma}S^{(i)}_{\delta n}=O(1/n).$

Combining with \threfgoodstartingstatesdistance,

d_{n}(\delta n+t)\leq\max_{\sigma\in\tilde{\Omega}}\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\mu\|_{TV}+O(1/n).

(4)

Definition ( $2m$ -coordinate chain).

Let $\tilde{\sigma}\in\Omega$ be a reference configuration. For $\sigma\in\Omega$ and each $i$ , define

	$\displaystyle U_{i}(\sigma)\colonequals\|\{v\in J_{i}:\sigma^{(i)}(v)=\tilde{\sigma}^{(i)}(v)=1\}\|,$
	$\displaystyle V_{i}(\sigma)\colonequals\|\{v\in J_{i}:\sigma^{(i)}(v)=\tilde{\sigma}^{(i)}(v)=-1\}\|.$

For a chain $({\sigma_{t}})$ with the starting configuration $\sigma_{0}\in\Omega$ , define the $2m$ -coordinate chain with respect to $\tilde{\sigma}$ by

\displaystyle\mathbf{U}_{t}\colonequals({U^{(1)}_{t}},{V^{(1)}_{t}},\dots,{U^{(m)}_{t}},{V^{(m)}_{t}})\colonequals(U_{1}({\sigma_{t}}),V_{1}({\sigma_{t}}),\dots,U_{m}({\sigma_{t}}),V_{m}({\sigma_{t}})).

It is easy to see that the $2m$ -coordinate chain is again a Markov chain in its state space $\mathcal{U}\subseteq\mathbb{N}^{2m}$ and determines the magnetization chain $({S^{(1)}_{t}},\dots,{S^{(m)}_{t}})$ through the relation ${S^{(i)}_{t}}=2({U^{(i)}_{t}}-V^{(i)}_{t})/n-(\tilde{u}_{i}-\tilde{v}_{i})/{n}$ for $i=1,\dots,m$ .

Symmetry gives us the following lemma which is an adaptation of Lemma 3.4 in [10].

Lemma 4.4.

\thlabel

totalvariationdistance Let $({\sigma_{t}})$ be a chain starting at $\sigma\in\Omega$ . Consider the corresponding $2m$ -coordinate chain starting at $\mathbf{u}\in\mathcal{U}$ . Then

\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\mu\|_{TV}=\|\mathbb{P}_{\mathbf{u}}(({U^{(1)}_{t}},{V^{(1)}_{t}},\dots,{U^{(m)}_{t}},{V^{(m)}_{t}})\in\cdot)-\nu\|_{TV}

where $\nu$ is the stationary distribution of the $2m$ -coordinate chain.

Proof.

Since $\mu(\sigma)=e^{\beta n\sum_{i\neq j}S^{(i)}(\sigma)S^{(j)}(\sigma)}/Z(\beta)$ , given the $2m$ -coordinate $\mathbf{u}^{\prime}\in\mathcal{U}$ , the conditional $\mu$ -probability of the configurations is equiprobable. In other words, $\mu(\cdot|\Omega(\mathbf{u}^{\prime}))$ is uniform where $\Omega(\mathbf{u}^{\prime})$ is the set of configurations having the $2m$ -coordinate $\mathbf{u}^{\prime}$ . Also, by symmetry,

\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot\ |\mathbf{U}_{t}=\mathbf{u}^{\prime})

is uniform over $\Omega(\mathbf{u}^{\prime})$ . Thus,

\displaystyle\mathbb{P}_{\sigma}({\sigma_{t}}=\eta)-\mu(\eta)=\sum_{\mathbf{u}^{\prime}\in\mathcal{U}}\frac{\mathbbm{1}\{\eta\in\Omega(\mathbf{u}^{\prime})\}}{|\Omega(\mathbf{u}^{\prime})|}\left(\mathbb{P}_{\mathbf{u}^{\prime}}\left(\mathbf{U}_{t}=\mathbf{u}^{\prime}\right)-\mu(\Omega(\mathbf{u}^{\prime}))\right).

Taking absolute values, applying the triangular inequality, summing over $\eta$ , and changing the order of summation shows

\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\mu\|_{TV}\leq\|\mathbb{P}_{\mathbf{u}}(({U^{(1)}_{t}},{V^{(1)}_{t}},\dots,{U^{(m)}_{t}},{V^{(m)}_{t}})\in\cdot)-\nu\|_{TV}.

The reverse inequality holds since the $2m$ -coordinate chain is a function of the original chain $({\sigma_{t}})$ . ∎

Remark.

This lemma lets us look at the $2m$ -coordinate chain instead of the original chain when considering the total variation distance.

Fix a good configuration $\tilde{\sigma}\in\tilde{\Omega}$ . Recall $\tau_{mag}$ defined in \threfmagcoupling. We use the following coupling after $\tau_{mag}$ , which is a generalization of Lemma 3.5 of [10].

Lemma 4.5 (Post magnetization coupling).

\thlabel

postmagcoupling Let $\tilde{\sigma}\in\tilde{\Omega}$ be a good configuration. Suppose that two configurations $\sigma_{0},\sigma_{0}^{\prime}$ satisfy $S^{(i)}(\sigma_{0})=S^{(i)}(\sigma_{0}^{\prime})$ for $i=1,\dots,m$ . With respect to the good configuration $\tilde{\sigma}$ , define

\displaystyle\Theta_{i}\colonequals\Big{\{}\sigma\in\Omega:\min\{U_{i}(\sigma),\tilde{u}_{i}-U_{i}(\sigma),V_{i}(\sigma),\tilde{v}_{i}-V_{i}(\sigma)\}\geq\frac{|J_{i}|}{16}\Big{\}},\enspace\Theta\colonequals\bigcap_{i=1}^{m}\Theta_{i}

for each $i$ . Then there exists a coupling $({\sigma_{t}},{\sigma^{\prime}_{t}})$ of the Glauber dynamics with starting states $(\sigma_{0},\sigma_{0}^{\prime})$ satisfying:

	$\displaystyle\mathrm{(i)}\;\mathbf{S}_{t}=\mathbf{S}_{t}^{\prime}\ \text{for all}\ t\geq 0$
	$\displaystyle\mathrm{(ii)}\;\text{If ${R^{(i)}_{t}}\colonequals{U^{\prime(i)}_{t}}-{U^{(i)}_{t}}$, then }\mathbb{E}_{\sigma_{0},\sigma_{0}^{\prime}}\left({R^{(i)}_{t+1}}-{R^{(i)}_{t}}\|{\sigma_{t}},{\sigma^{\prime}_{t}}\right)=\frac{-{R^{(i)}_{t}}}{n},$
	$\displaystyle\qquad i=1,\dots,m$
	$\displaystyle\mathrm{(iii)}\;\text{There exists $c>0$ not depending on $n$ such that on the event}\ \{{\sigma_{t}},{\sigma^{\prime}_{t}}\in\Theta\},$
	$\displaystyle\qquad\mathbb{P}_{\sigma_{0},\sigma_{0}^{\prime}}\left({R^{(i)}_{t+1}}-{R^{(i)}_{t}}\neq 0\|{\sigma_{t}},{\sigma^{\prime}_{t}}\right)\geq c>0\;\text{for all}\ i=1,\dots,m.$

Proof.

We inductively define the coupling. The random spin $S$ determined by the randomness $I$ and $U$ is

\displaystyle S=\sum_{i=1}^{m}(\mathbbm{1}_{I\in J_{i},\;U\leq r_{+}(\sum_{j\neq i}{S^{(j)}_{t}})}-\mathbbm{1}_{I\in J_{i},\;U>r_{+}(\sum_{j\neq i}{S^{(j)}_{t}})}).

Suppose that $({\sigma_{t}},{\sigma^{\prime}_{t}})$ is given such that the statements hold for some $t\geq 0$ . Let $\sigma_{t+1}$ be determined $I$ and $U$ . If $I\in J_{i}$ for some $i$ , then choose $I^{\prime}$ randomly from $\{v\in J_{i}^{\prime}:{\sigma^{\prime}_{t}}(v)={\sigma_{t}}(I)\}$ . Update the primed chain by

\sigma_{t+1}^{\prime}(v)=\begin{cases}{\sigma^{\prime}_{t}}(v)&\text{if $v\neq I^{\prime}$}\\ S&\text{if $v=I^{\prime}$}\end{cases}\quad.

By the induction hypothesis $\mathbf{S}_{t}=\mathbf{S}_{t}^{\prime}$ , we have $\{v\in J_{i}^{\prime}:{\sigma^{\prime}_{t}}(v)={\sigma_{t}}(I)\}\neq\emptyset$ and $({\sigma^{\prime}_{t}})$ satisfies the Glauber dynamics. Also, $\mathbf{S}_{t+1}=\mathbf{S}_{t+1}^{\prime}$ with this coupling.

For $i=1,\dots,m$ , put

	$\displaystyle A_{i}(\sigma)\colonequals\{v\in J_{i}:\sigma(v)=\tilde{\sigma}(v)=1\},$
	$\displaystyle B_{i}(\sigma)\colonequals\{v\in J_{i}:\sigma(v)=-1,\ \tilde{\sigma}(v)=1\},$
	$\displaystyle C_{i}(\sigma)\colonequals\{v\in J_{i}:\sigma(v)=1,\ \tilde{\sigma}(v)=-1\},$
	$\displaystyle D_{i}(\sigma)\colonequals\{v\in J_{i}:\sigma(v)=\tilde{\sigma}(v)=-1\},$

so $|A_{i}(\sigma)|=U_{i}(\sigma)$ , $|B_{i}(\sigma)|=\tilde{u}_{i}-U_{i}(\sigma)$ , $|C_{i}(\sigma)|=\tilde{v}_{i}-V_{i}(\sigma)$ , and $|D_{i}(\sigma)|=V_{i}(\sigma)$ .

Now we calculate ${R^{(1)}_{t+1}}-{R^{(i)}_{t}}$ with the above coupling. The following table shows the one-step dynamics of ${R^{(1)}_{t}}$ .

$I$	$I^{\prime}$	$S$	${R^{(1)}_{t+1}}-{R^{(1)}_{t}}$
$B_{1}({\sigma_{t}})$	$D_{1}({\sigma^{\prime}_{t}})$	1	-1
$C_{1}({\sigma_{t}})$	$A_{1}({\sigma^{\prime}_{t}})$	-1	-1
$A_{1}({\sigma_{t}})$	$C_{1}({\sigma^{\prime}_{t}})$	-1	1
$D_{1}({\sigma_{t}})$	$B_{1}({\sigma^{\prime}_{t}})$	1	1
otherwise	otherwise	otherwise	0

Since ${S^{(1)}_{t}}={S^{\prime(1)}_{t}}$ implies ${R^{(1)}_{t}}\equiv{U^{\prime(1)}_{t}}-{U^{(1)}_{t}}={V^{\prime(1)}_{t}}-{V^{(1)}_{t}}$ ,

	$\displaystyle\mathbb{P}_{\sigma_{0},\sigma_{0}^{\prime}}({R^{(1)}_{t+1}}-{R^{(1)}_{t}}=-1\|{\sigma_{t}},{\sigma^{\prime}_{t}})\equalscolon a({U^{(1)}_{t}},{V^{(1)}_{t}},U_{2,t},V_{2,t})$
	$\displaystyle=\frac{\tilde{u}_{1}-{U^{(1)}_{t}}}{n}\frac{{V^{\prime(1)}_{t}}}{\tilde{u}_{1}-{U^{\prime(1)}_{t}}+{V^{\prime(1)}_{t}}}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}})+\frac{\tilde{v}_{1}-{V^{(1)}_{t}}}{n}\frac{{U^{\prime(1)}_{t}}}{\tilde{v}_{1}-{V^{\prime(1)}_{t}}+{U^{\prime(1)}_{t}}}r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}})$
	$\displaystyle=\frac{\tilde{u}_{1}-{U^{(1)}_{t}}}{n}\frac{{V^{(1)}_{t}}+{R^{(1)}_{t}}}{\tilde{u}_{1}-{U^{(1)}_{t}}+{V^{(1)}_{t}}}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}})+\frac{\tilde{v}_{1}-{V^{(1)}_{t}}}{n}\frac{{U^{(1)}_{t}}+{R^{(1)}_{t}}}{\tilde{v}_{1}-{V^{(1)}_{t}}+{U^{(1)}_{t}}}r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}}).$

Likewise,

	$\displaystyle\mathbb{P}_{\sigma_{0},\sigma_{0}^{\prime}}({R^{(1)}_{t+1}}-{R^{(1)}_{t}}=1\|{\sigma_{t}},{\sigma^{\prime}_{t}})\equalscolon b({U^{(1)}_{t}},{V^{(1)}_{t}},U_{2,t},V_{2,t})$
	$\displaystyle=\frac{{U^{(1)}_{t}}}{n}\frac{\tilde{v}_{1}-{V^{\prime(1)}_{t}}}{{U^{\prime(1)}_{t}}+\tilde{v}_{1}-{V^{\prime(1)}_{t}}}r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}})+\frac{{V^{(1)}_{t}}}{n}\frac{\tilde{u}_{1}-{U^{\prime(1)}_{t}}}{\tilde{u}_{1}-{U^{\prime(1)}_{t}}+{V^{\prime(1)}_{t}}}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}})$
	$\displaystyle=\frac{{U^{(1)}_{t}}}{n}\frac{\tilde{v}_{1}-({V^{(1)}_{t}}+{R^{(1)}_{t}})}{{U^{(1)}_{t}}+\tilde{v}_{1}-{V^{(1)}_{t}}}r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}})+\frac{{V^{(1)}_{t}}}{n}\frac{\tilde{u}_{1}-({U^{(1)}_{t}}+{R^{(1)}_{t}})}{\tilde{u}_{1}-{U^{(1)}_{t}}+{V^{(1)}_{t}}}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}}).$

Thus, by a direct calculation,

	$\displaystyle\mathbb{E}_{\sigma_{0},\sigma_{0}^{\prime}}({R^{(1)}_{t+1}}-{R^{(1)}_{t}}\|{\sigma_{t}},{\sigma^{\prime}_{t}})=b-a$
	$\displaystyle=\frac{-{R^{(1)}_{t}}}{n}\biggl{(}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}})+r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}})\biggr{)}=\frac{-{R^{(1)}_{t}}}{n}.$

Moreover, on the event $\{{\sigma_{t}},{\sigma^{\prime}_{t}}\in\Theta\}$ , $(\tilde{u}_{1},\tilde{v}_{1},\dots,\tilde{u}_{m},\tilde{v}_{m})\in\tilde{\Lambda}$ implies ${U^{(1)}_{t}}\leq\tilde{u}_{1}-|J_{1}|/16\leq 3|J_{1}|/4-|J_{1}|/16=11|J_{1}|/16$ , and $\tilde{u}_{1}-{U^{(1)}_{t}}\leq 3|J_{1}|/4-|J_{1}|/16=11|J_{1}|/16$ . The same upper bound holds for $\tilde{v}_{1}-{V^{(1)}_{t}}$ and ${V^{(1)}_{t}}$ . Thus, on the event $\{{\sigma_{t}},{\sigma^{\prime}_{t}}\in\Theta\}$ ,

	$\displaystyle\mathbb{P}_{\sigma_{0},\sigma_{0}^{\prime}}({R^{(1)}_{t+1}}-{R^{(1)}_{t}}\neq 0\|{\sigma_{t}},{\sigma^{\prime}_{t}})$	$\displaystyle\geq b\geq\frac{p_{1}}{16}\frac{\frac{1}{16}r_{-}(\sum_{j\neq 1}{S^{(j)}_{t}})}{\frac{11}{16}+\frac{11}{16}}+\frac{p_{1}}{16}\frac{\frac{1}{16}r_{+}(\sum_{j\neq 1}{S^{(j)}_{t}})}{\frac{11}{16}+\frac{11}{16}}$
		$\displaystyle=\frac{p_{1}}{352}.$

Similarly, for $i>1$ , $\mathbb{P}_{\sigma_{0},\sigma_{0}^{\prime}}({R^{(i)}_{t+1}}-{R^{(i)}_{t}}\neq 0|{\sigma_{t}},{\sigma^{\prime}_{t}})\geq{p_{i}}/{352}\geq{p_{1}}/{352}>0$ , which concludes the induction. ∎

5. Upper and Lower Bounds in the high temperature regime

5.1. Upper Bound

Theorem 5.1.

\thlabel

upperbound For $\beta<\beta_{cr}$ , we have

\lim_{\gamma\to\infty}\limsup_{n\to\infty}d_{n}(t_{n}+\gamma n)=0.

Proof.

Let $\nu$ be the stationary measure for the $2m$ -coordinate chain. For any $A\subseteq\mathcal{U}$ ,

	$\displaystyle\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in A)-\nu(A)\|$	$\displaystyle=\Big{\|}\sum_{\mathbf{u}^{\prime}\in\mathcal{U}}\nu(\mathbf{u}^{\prime})\left(\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in A)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in A)\right)\Big{\|}$
		$\displaystyle\leq\sum_{\mathbf{u}^{\prime}\in\mathcal{U}}\nu(\mathbf{u}^{\prime})\\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in\cdot)\\|_{TV}$
		$\displaystyle\leq\max_{\mathbf{u}^{\prime}\in\mathcal{U}}\\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in\cdot)\\|_{TV}.$

Thus, taking supremum over $A\subseteq\mathcal{U}$ and $\mathbf{u}\in\tilde{\Lambda}$ ,

\displaystyle\max_{\mathbf{u}\in\tilde{\Lambda}}\|\mathbb{P}_{\mathbf{u}}\left(\mathbf{U}_{t}\in\cdot\right)-\nu\|_{TV}\leq\max_{\begin{subarray}{c}\mathbf{u}\in\tilde{\Lambda},\\ \mathbf{u}^{\prime}\in\mathcal{U}\end{subarray}}\|\mathbb{P}_{\mathbf{u}}\left(\mathbf{U}_{t}\in\cdot\right)-\mathbb{P}_{\mathbf{u}^{\prime}}\left(\mathbf{U}_{t}^{\prime}\in\cdot\right)\|_{TV}.

Also, from inequality (4) and \threftotalvariationdistance,

	$\displaystyle d_{n}(\delta n+t)$	$\displaystyle\leq\max_{\sigma\in\tilde{\Omega}}\\|\mathbb{P}_{\sigma}({\sigma_{t}}\in\cdot)-\mu\\|+O(1/n)$
		$\displaystyle=\max_{\mathbf{u}\in\tilde{\Lambda}}\\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\nu\\|_{TV}+O(1/n).$

For $2m$ -coordinate chains $\mathbf{U}_{t}$ and $\mathbf{U}_{t}^{\prime}$ with respect to a fixed $\tilde{\sigma}\in\tilde{\Omega}$ starting at $\mathbf{u}\in\mathcal{U}$ and $\mathbf{u}^{\prime}\in\mathcal{U}$ , respectively, put

\tau_{tot,c}\colonequals\min\{t\geq 0:\mathbf{U}_{t}=\mathbf{U}_{t}^{\prime}\}.

It is a standard fact [11, Section 5.2] that

\displaystyle\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in\cdot)\|_{TV}\leq\mathbb{P}_{\mathbf{u},\mathbf{u}^{\prime}}(\tau_{tot,c}>t).

Combining all the above results, it suffices to bound

\max_{\begin{subarray}{c}\mathbf{u}\in\tilde{\Lambda},\\ \mathbf{u}^{\prime}\in\mathcal{U}\end{subarray}}\mathbb{P}_{\mathbf{u},\mathbf{u}^{\prime}}(\tau_{tot,c}>t).

With the above considerations, fix a good starting configuration $\tilde{\sigma}\in\tilde{\Omega}$ with the associated $2m$ -coordinates $\tilde{\mathbf{u}}=(\tilde{u}_{1},\tilde{v}_{1},\dots,\tilde{u}_{m},\tilde{v}_{m})\in\tilde{\Lambda}$ and an arbitrary starting configuration $\sigma^{\prime}\in\Omega$ . Put

t_{n}(\gamma)\colonequals t_{n}+\gamma n,\enspace H_{M}\colonequals\{\tau_{mag}\leq t_{n}(\gamma)\}.

The first step is the magnetization coupling phase. By \threfmagcoupling, there exists a coupling $({\sigma_{t}},{\sigma^{\prime}_{t}})$ for $t\leq t_{n}(\gamma)$ with starting configurations $(\tilde{\sigma},\sigma^{\prime})$ such that

\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{M}^{c})\leq O(1/\sqrt{\gamma}).

The next step is the $2m$ -coordinate chain coupling phase. For $i=1,\dots,m$ , define

	$\displaystyle\tau_{i,c}\colonequals\min\{t\geq 0:({U^{(i)}_{t}},V^{(i)}_{t})=({U^{\prime(i)}_{t}},V^{\prime(i)}_{t})\},$
	$\displaystyle\Theta_{i}\colonequals\Big{\{}\sigma\in\Omega:\min\{U_{i}(\sigma),\tilde{u}_{i}-U_{i}(\sigma),V_{i}(\sigma),\tilde{v}_{i}-V_{i}(\sigma)\}\geq\frac{\|J_{i}\|}{16}\Big{\}},$
	$\displaystyle H_{i}(t)\colonequals\{{\sigma^{(i)}_{t}},{\sigma^{\prime(i)}_{t}}\in\Theta_{i}\},\quad H_{i}\colonequals\bigcap_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}H_{i}(t),\quad H_{tot}\colonequals\bigcap_{i=1}^{m}H_{i}.$

We have defined the two coordinate chains with respect to $\tilde{\sigma}$ . On the event $H_{M}$ , for $t\geq t_{n}(\gamma)$ , we use the coupling in \threfpostmagcoupling, while on the event $H_{M}^{c}$ , we let the chains run independently for $t\geq t_{n}(\gamma)$ since we do not care about this un-probable event.

Our first claim is that

\displaystyle\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{i}^{c})\leq\gamma O(1/n),\enspace i=1,\dots,m.

To that end, observe that

\begin{split}\{{\sigma^{(i)}_{t}}\notin\Theta_{i}\}\subseteq\{{U^{(i)}_{t}}<|J_{i}|/16\}&\cup\{\tilde{u}_{i}-{U^{(i)}_{t}}<|J_{i}|/16\}\\ &\cup\{V^{(i)}_{t}<|J_{i}|/16\}\cup\{\tilde{v}_{i}-V^{(i)}_{t}<|J_{i}|/16\}.\end{split}

Notice $\tilde{u}_{i}\geq|J_{i}|/4$ implies

	$\displaystyle\{{U^{(i)}_{t}}<\|J_{i}\|/16\}$	$\displaystyle\subseteq\{\tilde{u}_{i}-{U^{(i)}_{t}}>3\|J_{i}\|/16\},$
	$\displaystyle\{\tilde{u}_{i}-{U^{(i)}_{t}}<\|J_{i}\|/16\}$	$\displaystyle\subseteq\{{U^{(i)}_{t}}>3\|J_{i}\|/16\}.$

Similarly, $\tilde{v}_{i}\geq|J_{i}|/4$ implies

\begin{split}\{V^{(i)}_{t}<|J_{i}|/16\}&\subseteq\{\tilde{v}_{i}-V^{(i)}_{t}>3|J_{i}|/16\},\\ \{\tilde{v}_{i}-V^{(i)}_{t}<|J_{i}|/16\}&\subseteq\{V^{(i)}_{t}>3|J_{i}|/16\}.\end{split}

Put

\tilde{A}_{i}\colonequals\{k\in J_{i}:\tilde{\sigma}(k)=1\},\enspace i=1,\dots,m.

Then, following the notation in \threfexpectedmagbound, $|M_{t}(\tilde{A}_{i})|=|{U^{(i)}_{t}}-(\tilde{u}_{i}-{U^{(i)}_{t}})|$ implies

\{{U^{(i)}_{t}}<|J_{i}|/16\}\cup\{\tilde{u}_{i}-{U^{(i)}_{t}}<|J_{i}|/16\}\subseteq\{|M_{t}(\tilde{A}_{i})|\geq|J_{i}|/8\}.

Similarly, $|M_{t}(J_{i}\setminus\tilde{A}_{i})|=|V^{(i)}_{t}-(\tilde{v}_{i}-V^{(i)}_{t})|$ implies

\{V^{(i)}_{t}<|J_{i}|/16\}\cup\{\tilde{v}_{i}-V^{(i)}_{t}<|J_{i}|/16\}\subseteq\{|M_{t}(J_{i}\setminus\tilde{A}_{i})|\geq|J_{i}|/8\}.

Combining all the above results, we obtain

\{{\sigma^{(i)}_{t}}\notin\Theta_{i}\}\subseteq\{|M_{t}(\tilde{A}_{i})|\geq|J_{i}|/8\}\cup\{|M_{t}(J_{i}\setminus\tilde{A}_{i})|\geq|J_{i}|/8\}.

A parallel argument for the primed chain shows

\{{\sigma^{\prime(i)}_{t}}\notin\Theta_{i}\}\subseteq\{|M_{t}^{\prime}(\tilde{A}_{i})|\geq|J_{i}|/8\}\cup\{|M_{t}^{\prime}(J_{i}\setminus\tilde{A}_{i})|\geq|J_{i}|/8\}.

In conclusion,

	$\displaystyle H_{i}(t)^{c}$	$\displaystyle=\{{\sigma^{(i)}_{t}}\notin\Theta_{i}\}\cup\{{\sigma^{\prime(i)}_{t}}\notin\Theta_{i}\}$
		$\displaystyle\subseteq\{\|M_{t}(\tilde{A}_{i})\|\geq\|J_{i}\|/8\}\cup\{\|M_{t}(J_{i}\setminus\tilde{A}_{i})\|\geq\|J_{i}\|/8\}$
		$\displaystyle\qquad\qquad\qquad\qquad\quad\enspace\;\cup\{\|M_{t}^{\prime}(\tilde{A}_{i})\|\geq\|J_{i}\|/8\}\cup\{\|M_{t}^{\prime}(J_{i}\setminus\tilde{A}_{i})\|\geq\|J_{i}\|/8\}.$

Define

B\colonequals\bigcup_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}\{|M_{t}(\tilde{A}_{i})|\geq|J_{i}|/8\},\quad Y\colonequals\sum_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}\mathbbm{1}_{\{|M_{t}(\tilde{A}_{i})|\geq|J_{i}|/16\}}.

Since $M_{t}(\tilde{A}_{i})$ has increments in $\{-1,0,1\}$ , we have $B\subseteq\{Y\geq|J_{i}|/16\}$ . By Chebyshev’s inequality, $\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(B)\leq c\mathbb{E}_{\tilde{\sigma},\sigma^{\prime}}(Y)/n$ for some constant $c>0$ . From \threfexpectedmagbound, for $t\geq t_{n}$ , $\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(|M_{t}(\tilde{A}_{i})|\geq|J_{i}|/16)=O(1/n)$ , so $\mathbb{E}_{\tilde{\sigma},\sigma^{\prime}}(Y)=\gamma O(1)$ . Thus, $\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(B)=\gamma O(1/n)$ . Similar results hold for $\bigcup_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}\{|M_{t}(J_{i}\setminus\tilde{A}_{i})|\geq|J_{i}|/8\}$ , $\bigcup_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}\{|M_{t}^{\prime}(\tilde{A}_{i})|\geq|J_{i}|/8\}$ , and $\bigcup_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}\{|M_{t}^{\prime}(J_{i}\setminus\tilde{A}_{i})|\geq|J_{i}|/8\}$ . In conclusion,

\displaystyle\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{i}^{c})=\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}\Biggl{(}\bigcup_{t\in[t_{n}(\gamma),t_{n}(2\gamma)]}H_{i}(t)^{c}\Biggr{)}\leq 4\gamma O(1/n),

which proves our first claim.

From the first claim,

\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{tot}^{c})\leq\sum_{i=1}^{m}\mathbb{P}_{\sigma,\sigma^{\prime}}(H_{i}^{c})=\gamma O(1/n).

Now, condition on the event $H_{M}$ . Recalling the fact that \threfpostmagcoupling assures $\mathbf{S}_{t}=\mathbf{S}_{t}^{\prime}$ for $t\geq t_{n}(\gamma)$ on the event $H_{M}$ , we can make ${R^{(i)}_{t}}$ stay zero after $\tau_{i,c}$ , using the modified monotone update on $J_{i}$ whenever a site in $J_{i}$ is chosen to be updated. Thus, on $H_{M}$ ,

\tau_{tot,c}=\max_{1\leq i\leq m}\tau_{i,c}.

Our second claim is that

\displaystyle\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{i,c}>t_{n}(2\gamma),H_{i},H_{M})=O(1/\sqrt{\gamma}),\enspace i=1,\dots,m.

From \threfsupermartingale and \threfpostmagcoupling, $\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{i,c}>t_{n}(2\gamma),H_{i},H_{M}|\sigma_{t_{n}(\gamma)},\sigma_{t_{n}(\gamma)}^{\prime})\leq{c|R^{(i)}_{t_{n}(\gamma)}|}/{\sqrt{n\gamma}}$ for some $c>0$ . Taking expectation yields,

\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{i,c}>t_{n}(2\gamma),H_{i},H_{M})\leq\frac{c\mathbb{E}_{\tilde{\sigma},\sigma^{\prime}}|R^{(i)}_{t_{n}(\gamma)}|}{\sqrt{n\gamma}}

However, for any $t>0$ , $|{R^{(i)}_{t}}|=|U_{t}^{\prime}-U_{t}|=|M_{t}^{\prime}(\tilde{A}_{i})-M_{t}(\tilde{A}_{i})|$ , so from \threfexpectedmagbound, $\mathbb{E}_{\tilde{\sigma},\sigma^{\prime}}|R^{(i)}_{t_{n}(\gamma)}|\leq\mathbb{E}_{\sigma^{\prime}}|M_{t_{n}(\gamma)}^{\prime}(\tilde{A}_{i})|+\mathbb{E}_{\tilde{\sigma}}|M_{t_{n}(\gamma)}(\tilde{A}_{i})|=O(\sqrt{n})$ , which proves our second claim.

From the second claim,

	$\displaystyle\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{tot,c}>t_{n}(2\gamma),H_{tot},H_{M})\leq\sum_{i=1}^{m}\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{i,c}>t_{n}(2\gamma),H_{tot},H_{M})$
	$\displaystyle\leq\sum_{i=1}^{m}\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{i,c}>t_{n}(2\gamma),H_{i},H_{M})=O(1/\sqrt{\gamma}).$

Combining all the above results,

	$\displaystyle\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{tot,c}>t_{n}(2\gamma))$
	$\displaystyle\leq\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(\tau_{tot,c}>t_{n}(2\gamma),H_{tot},H_{M})+\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{tot}^{c})+\mathbb{P}_{\tilde{\sigma},\sigma^{\prime}}(H_{M}^{c})$
	$\displaystyle=O(1/\sqrt{\gamma})+\gamma O(1/n)+O(1/\sqrt{\gamma}).$

Finally,

\displaystyle d_{n}(t_{n}+(2\gamma+\delta)n)\leq O(1/\sqrt{\gamma})+\gamma O(1/n)+O(1/n),

which gives us the result upon taking limits. ∎

5.2. Lower Bound

We first analyze the drift of magnetization chains. Let $1\leq i\leq m$ and $\mathcal{F}_{t}$ be the $\sigma$ -algebra generated by ${S^{(1)}_{t}},\dots,{S^{(m)}_{t}}$ . By a direct calculation,

$\displaystyle\mathbb{E}[{S^{(i)}_{t+1}}-{S^{(i)}_{t}}\|\mathcal{F}_{t}]$	$\displaystyle=\frac{2}{n}p_{i}\frac{\|J_{i}\|-n{S^{(i)}_{t}}}{2\|J_{i}\|}r_{+}(\sum_{j\neq i}{S^{(j)}_{t}})-\frac{2}{n}p_{i}\frac{\|J_{i}\|+n{S^{(i)}_{t}}}{2\|J_{i}\|}r_{-}(\sum_{j\neq i}{S^{(j)}_{t}})$
	$\displaystyle=\frac{2}{n}\frac{p_{i}-{S^{(i)}_{t}}}{2}r_{+}(\sum_{j\neq i}{S^{(j)}_{t}})-\frac{2}{n}\frac{p_{i}+{S^{(i)}_{t}}}{2}r_{-}(\sum_{j\neq i}{S^{(j)}_{t}})$
	$\displaystyle=\frac{1}{n}\biggl{(}-{S^{(i)}_{t}}+p_{i}\tanh({\beta}\sum_{j\neq i}{S^{(j)}_{t}})\biggr{)}.$	(5)

The following simple lemma is the main tool to get the lower bound in \threflowerbound.

Lemma 5.2 (Proposition 7.9, [11]).

\thlabel

statistics Let $f\colon\mathcal{S}\to\mathbb{R}$ be a measurable function and $\nu_{1}$ , $\nu_{2}$ be two probability measures on $\mathcal{S}$ . Let $\sigma_{*}^{2}\colonequals\max\{\mathbb{V}\mathrm{ar}_{\nu_{1}}f,\ \mathbb{V}\mathrm{ar}_{\nu_{2}}f\}$ . If $|\mathbb{E}_{\nu_{1}}f-\mathbb{E}_{\nu_{2}}f|\geq r\sigma_{*}$ , then

\|\nu_{1}-\nu_{2}\|_{TV}\geq 1-\frac{8}{r^{2}}

Positive starting configurations give us the following result.

Lemma 5.3.

\thlabel

finallemma Let $\mathbf{s}\geq\mathbf{0}$ be the starting magentization. Then, for $t\geq 0$ ,

\mathbb{E}_{\mathbf{s}}\|\mathbf{S}_{t}\|_{1}\leq g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2}+O(1/\sqrt{n}).

Proof.

Consider the case that $|J_{i}|$ is odd for each $i=1,\dots,m$ . Let $\nu$ be the starting distribution such that $\mathbf{s}_{+}^{\prime}=(\frac{1}{n},\dots,\frac{1}{n})$ with probability $\frac{1}{2}$ and $\mathbf{s}_{-}^{\prime}=(-\frac{1}{n},\dots,-\frac{1}{n})$ with probability $\frac{1}{2}$ .

By \threfgeneralcontraction, since $\mathbf{s}\geq\mathbf{s}_{+}^{\prime}$ in this case,

\displaystyle\mathbf{0}

\displaystyle\leq\mathbb{E}_{\mathbf{s},\nu}(\mathbf{S}_{t}-\mathbf{S}_{t}^{\prime})\leq\frac{1}{2}\mathbf{A}^{t}(\mathbf{s}-\mathbf{s}_{+}^{\prime})+\frac{1}{2}\mathbf{A}^{t}(\mathbf{s}-\mathbf{s}_{-}^{\prime})=\mathbf{A}^{t}\mathbf{s}.

However, $\mathbb{E}_{\nu}{S^{\prime(i)}_{t}}=0$ for $i=1,\dots,m$ by the remark after \threfmagmarkov. Thus, $\mathbf{0}\leq\mathbb{E}_{\mathbf{s}}\mathbf{S}_{t}\leq\mathbf{A}^{t}\mathbf{s}$ , so by \threfnewlemma,

\displaystyle 0\leq\sum_{i=1}^{m}\mathbb{E}_{\mathbf{s}}{S^{(i)}_{t}}\leq\|\mathbf{A}^{t}\mathbf{s}\|_{1}\leq g^{t}\left(\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\right)^{1/2}.

From \threfmagvariancebound and Cauchy-Schwartz inequality, since $0\leq\mathbb{E}_{\mathbf{s}}{S^{(i)}_{t}}$ for $i=1,\dots,m$ ,

	$\displaystyle\mathbb{E}_{\mathbf{s}}\\|\mathbf{S}_{t}\\|_{1}=\sum_{i=1}^{m}\mathbb{E}_{\mathbf{s}}\|{S^{(i)}_{t}}\|\leq\sum_{i=1}^{m}\left(\|\mathbb{E}_{\mathbf{s}}{S^{(i)}_{t}}\|+\sqrt{\mathbb{V}\mathrm{ar}_{\mathbf{s}}{S^{(i)}_{t}}}\right)=\sum_{i=1}^{m}\mathbb{E}_{\mathbf{s}}{S^{(i)}_{t}}+\sum_{i=1}^{m}\sqrt{\mathbb{V}\mathrm{ar}_{\mathbf{s}}{S^{(i)}_{t}}}$
	$\displaystyle\leq g^{t}\Biggl{(}\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\Biggr{)}^{1/2}+\Biggl{(}m\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}\Biggr{)}^{1/2}=g^{t}\Biggl{(}\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}\Biggr{)}^{1/2}+O(\frac{1}{\sqrt{n}}).$

Other cases of $|J_{i}|$ can similarly be shown by considering $0$ instead of $\frac{1}{n}$ whenever the partition has even number of sites. ∎

Finally, we prove the lower bound.

Theorem 5.4.

\thlabel

lowerbound For $\beta<\beta_{cr}$ , we have

\lim_{\gamma\to\infty}\liminf_{n\to\infty}d_{n}(t_{n}-\gamma n)=1.

Proof.

Since the magnetization chain is a projection of the original chain, it suffices to provide a lower bound on the total variation norm of the magnetization chain. Using $\tanh x\geq x-x^{2}/3$ for $x\in\mathbb{R}$ , from equations (5), we have

\displaystyle\mathbb{E}({S^{(i)}_{t+1}}|\mathcal{F}_{t})

\displaystyle\geq(1-\frac{1}{n}){S^{(i)}_{t}}+\frac{p_{i}}{n}\Biggl{(}\beta\sum_{j\neq i}{S^{(j)}_{t}}-\frac{\beta^{2}(\sum_{j\neq i}{S^{(j)}_{t}})^{2}}{3}\Biggr{)}

for each $i=1,\dots,m$ . In the matrix form,

\displaystyle\mathbb{E}(\mathbf{S}_{t+1}|\mathcal{F}_{t})

\displaystyle\geq\mathbf{A}\mathbf{S}_{t}-\mathbf{x}

where $\mathbf{x}=\frac{\beta^{2}}{3n}(p_{1}(\sum_{j\neq 1}{S^{(j)}_{t}})^{2},\dots,p_{m}(\sum_{j\neq m}{S^{(j)}_{t}})^{2})^{T}$ . Recall the definition of $\mathbf{a}^{T}\colonequals(a_{1},\dots,a_{m})>\mathbf{0}$ with $\|\mathbf{a}\|_{1}=1$ being the left eigenvector of $\mathbf{A}$ with eigenvalue $g$ . Then $\mathbb{E}(\mathbf{a}^{T}\mathbf{S}_{t+1}|\mathcal{F}_{t})\geq\mathbf{a}^{T}\mathbf{A}\mathbf{S}_{t}-\mathbf{a}^{T}\mathbf{x}=g\mathbf{a}^{T}\mathbf{S}_{t}-\mathbf{a}^{T}\mathbf{x}$ , i.e.,

\mathbb{E}\Bigl{(}\sum_{i=1}^{m}a_{i}{S^{(i)}_{t+1}}|\mathcal{F}_{t}\Bigr{)}\geq g\sum_{i=1}^{m}a_{i}{S^{(i)}_{t}}-\frac{\beta^{2}}{3n}\sum_{i=1}^{m}a_{i}p_{i}\biggl{(}\sum_{j\neq i}{S^{(j)}_{t}}\biggr{)}^{2}.

(6)

Observe that

\displaystyle\sum_{i=1}^{m}a_{i}p_{i}\biggl{(}\sum_{j\neq i}{S^{(j)}_{t}}\biggr{)}^{2}\leq\sum_{k=1}^{m}a_{k}p_{k}\bigg{(}\sum_{j=1}^{m}|S^{(j)}_{t}|\bigg{)}^{2}=\biggl{(}\sum_{k=1}^{m}a_{k}p_{k}\biggr{)}\|\mathbf{S}_{t}\|_{1}^{2}.

Thus, upon taking expectation in equation (6),

\displaystyle\mathbb{E}\left(\sum_{i=1}^{m}a_{i}{S^{(i)}_{t+1}}\right)\geq g\mathbb{E}\left(\sum_{i=1}^{m}a_{i}{S^{(i)}_{t}}\right)-\frac{\beta^{2}}{3n}\left(\sum_{i=1}^{m}a_{i}p_{i}\right)\mathbb{E}\|\mathbf{S}_{t}\|_{1}^{2}.

We claim that,

\displaystyle\mathbb{E}\|\mathbf{S}_{t}\|_{1}^{2}\leq(\mathbb{E}\|\mathbf{S}_{t}\|_{1})^{2}+O(1/n).

Since $\mathbb{E}\|\mathbf{S}_{t}\|_{1}^{2}=(\mathbb{E}\|\mathbf{S}_{t}\|_{1})^{2}+\mathbb{V}\mathrm{ar}\|\mathbf{S}_{t}\|_{1}$ , it suffices to show $\mathbb{V}\mathrm{ar}\|\mathbf{S}_{t}\|_{1}\leq O(1/n)$ . However, from \threfmagvariancebound,

	$\displaystyle\mathbb{V}\mathrm{ar}\\|\mathbf{S}_{t}\\|_{1}$	$\displaystyle=\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}\|{S^{(i)}_{t}}\|+2\sum_{i>j}\mathrm{Cov}(\|{S^{(i)}_{t}}\|,\|{S^{(j)}_{t}}\|)$
		$\displaystyle\leq\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}+2\sum_{i>j}\sqrt{\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}}\sqrt{\mathbb{V}\mathrm{ar}{S^{(j)}_{t}}}$
		$\displaystyle\leq\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}+\sum_{i>j}(\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}+\mathbb{V}\mathrm{ar}{S^{(j)}_{t}})=m\sum_{i=1}^{m}\mathbb{V}\mathrm{ar}{S^{(i)}_{t}}=O(1/n),$

which proves the claim.

Put $Z_{t}\colonequals\sum_{i=1}^{m}a_{i}{S^{(i)}_{t}}/g^{t}$ . Then, from the claim above,

\displaystyle\mathbb{E}Z_{t+1}-\mathbb{E}Z_{t}\geq-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3ng^{t+1}}\left((\mathbb{E}\|\mathbf{S}_{t}\|_{1})^{2}+O(1/n)\right).

Assume that $\mathbf{s}\geq\mathbf{0}$ is a non-negative starting magnetization. Recalling the definition $\upsilon\colonequals n(1-g)$ , from \threffinallemma and the fact $\sum_{i}{(s^{(i)})^{2}}/{p_{i}}\leq 1$ ,

	$\displaystyle\mathbb{E}_{\mathbf{s}}Z_{t+1}-\mathbb{E}_{\mathbf{s}}Z_{t}$	$\displaystyle\geq-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3ng^{t+1}}\Biggl{(}\biggl{(}g^{t}\biggl{(}\sum_{i}{(s^{(i)})^{2}}/{p_{i}}\biggr{)}^{1/2}+O(1/\sqrt{n})\biggr{)}^{2}+O(1/n)\Biggr{)}$
		$\displaystyle\geq-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3(n-\upsilon)}\Biggl{(}g^{t}\sum_{i}{(s^{(i)})^{2}}/{p_{i}}+O(1/\sqrt{n})+\frac{1}{g^{t}}O(1/n)\Biggr{)}.$

Iterating from $0$ to $t-1$ ,

	$\displaystyle\mathbb{E}_{\mathbf{s}}Z_{t}-Z_{0}$	$\displaystyle\geq-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3(n-\upsilon)}\left(\frac{1-g^{t}}{\upsilon/n}\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}+tO(1/\sqrt{n})+\frac{n-\upsilon}{\upsilon}(\frac{1}{g^{t}}-1)O(1/n)\right)$
		$\displaystyle=-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3\upsilon(1-\upsilon/n)}(1-g^{t})\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3(n-\upsilon)}tO(1/\sqrt{n})$
		$\displaystyle\quad\;-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3\upsilon}(\frac{1}{g^{t}}-1)O(1/n).$

For brevity, let us prefer to use $\upsilon$ rather than use $\beta_{cr}$ in view of \threfupsilon. Consider the step $t_{*}\colonequals t_{n}-\gamma n/\upsilon=\frac{1}{2\upsilon}n\ln n-\frac{\gamma n}{\upsilon}$ . Observe that $1-1/x\geq e^{-1/(x-1)}$ for $x>1$ implies

g^{t_{*}}\geq\frac{e^{\gamma}}{n^{n/(2(n-\upsilon))}}.

Then

	$\displaystyle\mathbb{E}_{\mathbf{s}}Z_{t_{*}}-\sum_{i=1}^{m}a_{i}s_{i}\geq$	$\displaystyle-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3\upsilon(1-{\upsilon}/{n})}\left(1-\frac{e^{\gamma}}{n^{{n}/{(2(n-\upsilon))}}}\right)\sum_{i=1}^{m}\frac{(s^{(i)})^{2}}{p_{i}}$
		$\displaystyle-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3(n-\upsilon)}\left(\frac{1}{2\upsilon}n\ln n-\frac{\gamma n}{\upsilon}\right)O({1}/{\sqrt{n}})$
		$\displaystyle-\frac{\beta^{2}\sum_{i}a_{i}p_{i}}{3\upsilon}\left(\frac{n^{n/(2(n-\upsilon))}}{e^{\gamma}}-1\right)O({1}/{n}).$

The right-hand side of the above inequality converges to $-\frac{\beta^{2}\sum_{i}a_{i}p_{i}\sum_{i}{(s^{(i)})^{2}}/{p_{i}}}{3\upsilon}$ as $n\to\infty$ for every $\gamma>0$ .

We claim that if $n$ is large enough, then there exists $\mathbf{s}>\mathbf{0}$ such that

\sum_{i=1}^{m}a_{i}s_{i}-\frac{\beta^{2}\sum_{i}a_{i}p_{i}\sum_{i}{(s^{(i)})^{2}}/{p_{i}}}{3\upsilon}>0.

Consider $\mathbf{s}=\zeta\mathbf{p}$ where $0<\zeta<1$ is a constant to be determined. We want to find $\zeta$ such that

\sum_{i=1}^{m}a_{i}p_{i}\zeta-\frac{\beta^{2}\sum_{i}a_{i}p_{i}\sum_{i}{(p_{i}\zeta)^{2}}/{p_{i}}}{3\upsilon}>0,

which is equivalent to

3\upsilon>\beta^{2}\zeta.

From \threfupsilon, $\upsilon>0$ , so $\frac{3\upsilon}{\beta^{2}}\mathbf{p}>\mathbf{s}>\mathbf{0}$ assures that the inequality in the claim holds, and such a positive magnetization $\mathbf{s}\in\mathcal{S}$ exists since $n$ is large and $0\leq\beta<\beta_{cr}$ (if $\beta=0$ , choose $\mathbf{s}=\mathbf{p}$ ).

By the last claim, for large $n$ , there exists $\mathbf{s}\in\mathcal{S}$ and $\varepsilon>0$ such that

\mathbb{E}_{\mathbf{s}}(\sum_{i=1}^{m}a_{i}S^{(i)}_{t_{*}})\geq 2\varepsilon g^{t_{*}}\geq 2\varepsilon\frac{e^{\gamma}}{n^{n/(2(n-\upsilon))}}\geq\varepsilon\frac{e^{\gamma}}{\sqrt{n}}.

\thref

magvariancebound and the Cauchy-Schwartz inequality imply $\mathbb{V}\mathrm{ar}(\sum_{i=1}^{m}a_{i}S^{(i)}_{t_{*}})=O(\frac{1}{n})$ as $n\to\infty$ . Thus, by \threfzerospin and \threfstatistics, for some $c>0$ ,

\displaystyle\lim_{\gamma\to\infty}\liminf_{n\to\infty}d_{n}(t_{n}-\frac{\gamma n}{\upsilon})\geq\lim_{\gamma\to\infty}1-\frac{c}{\varepsilon^{2}e^{2\gamma}}=1.

∎

6. Exponentially slow mixing in the low temperature regime

Using a standard bottleneck ratio argument, we can show that the mixing time for the Glauber dynamics is exponential in the low temperature regime. The bottleneck ratio is defined as

\Phi\colonequals\min_{A:\mu(A)\leq 1/2}\frac{\sum_{x\in A,y\notin A}\mu(x)P(x,y)}{\mu(A)}

where $P$ is the transition matrix of the Glauber dynamics. The bottleneck ratio gives a lower bound of the mixing time (see [11, Theorem 7.4]):

t_{\mathrm{mix}}\geq\frac{1}{4\Phi}.

We need another characterization of the critical temperature $\beta_{cr}$ .

Lemma 6.1.

\thlabel

slowmixinglemma We have that

\beta_{cr}=\frac{\sum_{i}a_{i}^{2}p_{i}}{(\sum_{i}a_{i}p_{i})^{2}-\sum_{i}a_{i}^{2}p_{i}^{2}}

Proof.

From $\mathbf{a}^{T}\mathbf{A}=g\mathbf{a}^{T}$ , equation (3), and \threfupsilon, we have

\sum_{i}a_{i}p_{i}=\Big{(}p_{k}+\frac{1}{\beta_{cr}}\Big{)}a_{k}

for each $k=1,\dots,m$ . Multiplying $a_{k}p_{k}$ to both sides and summing over $k$ yields the result. ∎

Proof of \threflowtempresult.

It suffices to show that $\Phi\leq c_{1}\exp(-c_{2}n)$ for some positive constants $c_{1},c_{2}>0$ . By symmetry of the Hamiltonian, we have that $\mu(A)\leq 1/2$ where $A\colonequals\{\sigma:\sum_{i}S^{(i)}(\sigma)>0\}$ . Since the only way to go from $A$ to $A^{c}$ is to go through $B\colonequals\{\sigma:|\sum_{i}S^{(i)}(\sigma)|\leq 1/n\}$ , it holds that

\sum_{x\in A,y\notin A}\mu(x)P(x,y)\leq\mu(B).

Note that for any $\sigma\in\Omega$ ,

\mu(\sigma)=\frac{\exp\bigg{(}\frac{\beta n}{2}\Big{(}\big{(}\sum_{i}S^{(i)}(\sigma)\big{)}^{2}-\sum_{i}\big{(}S^{(i)}(\sigma)\big{)}^{2}\Big{)}\bigg{)}}{Z(\beta)}.

By the Cauchy-Schwartz inequality,

\mu(B)\leq\binom{n}{\lceil n/2\rceil}\frac{\exp\Big{(}\frac{\beta n}{2}\big{(}1-\frac{1}{m}\big{)}\big{(}\frac{1}{n}\big{)}^{2}\Big{)}}{Z(\beta)}\lesssim\binom{n}{\lceil n/2\rceil}/Z(\beta)

where $\lesssim$ denotes that the inequality holds for sufficiently large $n$ up to a constant not depending on $n$ . Using Stirling’s formula,

\Phi\lesssim\frac{\exp(n\ln 2)}{Z(\beta)\mu(A)}.

Now, consider the configurations with exactly $k_{i}np_{i}$ many " $+$ " spins in $J_{i}$ where $1/2\leq k_{i}\leq 1$ for each $i=1,\dots,m$ and there exists at least one $i$ such that $1/2<k_{i}$ . These configurations are members of $A$ and there are at least $\prod_{i=1}^{m}\binom{np_{i}}{k_{i}np_{i}}$ many such configurations. Using Stirling’s formula again, we obtain

Z(\beta)\mu(A)\gtrsim\Bigg{(}\frac{1}{\prod_{i=1}^{m}(1-k_{i})^{p_{i}(1-k_{i})}k_{i}^{p_{i}k_{i}}}\Bigg{)}^{n}e^{\frac{\beta n}{2}\big{(}(\sum_{i}(2k_{i}-1)p_{i})^{2}-\sum_{i}(2k_{i}-1)^{2}p_{i}^{2}\big{)}}.

Define a function $f$ through the equation

e^{nf(k_{1},\dots,k_{m})}\colonequals\Bigg{(}\frac{1}{\prod_{i=1}^{m}(1-k_{i})^{p_{i}(1-k_{i})}k_{i}^{p_{i}k_{i}}}\Bigg{)}^{n}e^{\frac{\beta n}{2}\big{(}(\sum_{i}(2k_{i}-1)p_{i})^{2}-\sum_{i}(2k_{i}-1)^{2}p_{i}^{2}\big{)}}.

Put $(k_{1},\dots,k_{m})=(1/2,\dots,1/2)+\gamma(v_{1},\dots,v_{m})$ where $v_{i}\geq 0$ for each $i=1,\dots,m$ , $\gamma\in\mathbb{R}$ , and $\sum_{i}v_{i}^{2}\neq 0$ . Fixing $v_{i}$ ’s, we can regard $f$ as a one-variable function of $\gamma$ , say $f=f(\gamma)$ , and this is equivalent to fixing a direction in $\mathbb{R}^{m}$ . A little calculation shows that

	$\displaystyle f(\gamma)$	$\displaystyle=2\beta\gamma^{2}\bigg{(}\Big{(}\sum_{i}v_{i}p_{i}\Big{)}^{2}-\sum_{i}v_{i}^{2}p_{i}^{2}\bigg{)}$
		$\displaystyle\quad-\sum_{i}p_{i}\big{(}(1/2-\gamma v_{i})\ln(1/2-\gamma v_{i})+(1/2+\gamma v_{i})\ln(1/2+\gamma v_{i})\big{)}$
	$\displaystyle f^{\prime}(\gamma)$	$\displaystyle=4\beta\gamma\bigg{(}\Big{(}\sum_{i}v_{i}p_{i}\Big{)}^{2}-\sum_{i}v_{i}^{2}p_{i}^{2}\bigg{)}-\sum_{i}p_{i}v_{i}\big{(}-\ln(1/2-\gamma v_{i})+\ln(1/2+\gamma v_{i})\big{)}$
	$\displaystyle f^{\prime\prime}(\gamma)$	$\displaystyle=4\beta\bigg{(}\Big{(}\sum_{i}v_{i}p_{i}\Big{)}^{2}-\sum_{i}v_{i}^{2}p_{i}^{2}\bigg{)}-\sum_{i}p_{i}v_{i}^{2}\bigg{(}\frac{1}{1/2-\gamma v_{i}}+\frac{1}{1/2+\gamma v_{i}}\bigg{)}$

where ^′ denotes a differentiation in $\gamma$ . Note that $f(0)=\ln 2$ and $f^{\prime}(0)=0$ . Thus, it suffices to show that there is a direction $(v_{1},\dots,v_{m})$ such that $f^{\prime\prime}(0)>0$ . \threfslowmixinglemma shows that the direction $(v_{1},\dots,v_{m})=(a_{1},\dots,a_{m})$ satisfies $f^{\prime\prime}(0)>0$ whenever $\beta>\beta_{cr}$ , which completes the proof. ∎

Remark.

Combined with the non-exponential mixing time of $O(n\ln n)$ whenever $\beta<\beta_{cr}$ , the above proof shows that $\inf_{\mathbf{v}\geq\mathbf{0},\mathbf{v}\neq\mathbf{0}}\frac{\sum_{i}v_{i}^{2}p_{i}}{(\sum_{i}v_{i}p_{i})^{2}-\sum_{i}v_{i}^{2}p_{i}^{2}}$ is achieved with the direction $(v_{1},\dots,v_{m})=\mathbf{a}^{T}$ .

Acknowledgments.

The author would like to thank Professor Insuk Seo for introducing the problem and sharing his limitless insight through numerous discussions. The author also acknowledges an anonymous user at math.stackexchange.com ²²2https://math.stackexchange.com/q/3553425 for the main idea of the proof in \threfnewlemma. Finally, the author acknowledges the anonymous reviewers for their helpful comments and careful reading of the paper.

References

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[3] P. Cuff et al. “Glauber Dynamics for the Mean-Field Potts Model” In Journal of Statistical Physics 149.3, 2012, pp. 432–477
[4] Persi Diaconis and Mehrdad Shahshahani “Generating a random permutation with random transpositions” In Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57.2, 1981, pp. 159–179
[5] Jian Ding, Eyal Lubetzky and Yuval Peres “The mixing time evolution of Glauber Dynamics for the mean-field Ising Model” In Communications in Mathematical Physics 289.2, 2009, pp. 725–764
[6] Ignacio Gallo, Adriano Barra and Pierluigi Contucci “Parameter Evaluation of a Simple Mean-Field Model of Social Interaction” In Mathematical Models and Methods in Applied Sciences 19, 2008
[7] José C. Hernández, Yevgeniy Kovchegov and Peter T. Otto “The aggregate path coupling method for the Potts model on bipartite graph” In Journal of Mathematical Physics 58.2, 2017, pp. 023303 DOI: 10.1063/1.4976502
[8] J.M. Kincaid and E.G.D. Cohen “Phase diagrams of liquid helium mixtures and metamagnets: Experiment and mean field theory” In Physics Reports 22.2, 1975, pp. 57–143
[9] Holger Knöpfel, Matthias Löwe, Kristina Schubert and Arthur Sinulis “Fluctuation Results for General Block Spin Ising Models” In Journal of Statistical Physics 178.5, 2020, pp. 1175–1200
[10] David A. Levin, Malwina J. Luczak and Yuval Peres “Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability” In Probability Theory and Related Fields, 2010, pp. 146–223
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[12] Eyal Lubetzky and Allan Sly “Cutoff for the Ising model on the lattice” In Inventiones mathematicae 191.3 Springer ScienceBusiness Media LLC, 2012, pp. 719–755
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[16] Carl D. Meyer “Matrix Analysis and Applied Linear Algebra” USA: Society for IndustrialApplied Mathematics, 2000

	$\displaystyle\geq\mathbb{P}(Y_{tot,t+1}\neq Y_{tot,t},I\in L_{t}\|\mathcal{F}_{t})\geq\mathbb{P}(E\dot{\cup}F\|\mathcal{F}_{t})$
	$\displaystyle\geq\frac{\|L_{t}\|+\sum_{i\in L_{t}}{\sigma_{t}}(i)}{2n}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}$
	$\displaystyle\enspace+\frac{\|L_{t}\|-\sum_{i\in L_{t}}{\sigma_{t}}(i)}{2n}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}$
	$\displaystyle\geq p_{1}\biggl{(}\frac{1-\tanh(\beta(1-p_{1}))}{2}\biggr{)}^{2}>0.$

	$\displaystyle\mathbb{P}_{\sigma,\sigma^{\prime}}(\tau_{mag}>\tau+\gamma^{\prime}n\|\sigma_{\tau},\sigma_{\tau}^{\prime})$	$\displaystyle\leq\mathbb{P}_{\sigma,\sigma^{\prime}}(Y_{tot,\tau+\gamma^{\prime}n}\geq a_{m}\|\sigma_{\tau},\sigma_{\tau}^{\prime})$
		$\displaystyle\leq\mathbb{E}_{\sigma,\sigma^{\prime}}[Y_{tot,\tau+\gamma^{\prime}n}\|\sigma_{\tau},\sigma_{\tau}^{\prime}]/a_{m}$
		$\displaystyle\leq g^{\gamma^{\prime}n}Y_{tot,\tau}/a_{m}\leq g^{\gamma^{\prime}n}a_{1}m/a_{m}\leq e^{-\upsilon\gamma^{\prime}}a_{1}m/a_{m}.$

	$\displaystyle\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in A)-\nu(A)\|$	$\displaystyle=\Big{\|}\sum_{\mathbf{u}^{\prime}\in\mathcal{U}}\nu(\mathbf{u}^{\prime})\left(\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in A)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in A)\right)\Big{\|}$
		$\displaystyle\leq\sum_{\mathbf{u}^{\prime}\in\mathcal{U}}\nu(\mathbf{u}^{\prime})\\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in\cdot)\\|_{TV}$
		$\displaystyle\leq\max_{\mathbf{u}^{\prime}\in\mathcal{U}}\\|\mathbb{P}_{\mathbf{u}}(\mathbf{U}_{t}\in\cdot)-\mathbb{P}_{\mathbf{u}^{\prime}}(\mathbf{U}_{t}^{\prime}\in\cdot)\\|_{TV}.$

	$\displaystyle\{{U^{(i)}_{t}}<\|J_{i}\|/16\}$	$\displaystyle\subseteq\{\tilde{u}_{i}-{U^{(i)}_{t}}>3\|J_{i}\|/16\},$
	$\displaystyle\{\tilde{u}_{i}-{U^{(i)}_{t}}<\|J_{i}\|/16\}$	$\displaystyle\subseteq\{{U^{(i)}_{t}}>3\|J_{i}\|/16\}.$

Cutoff phenomenon of the Glauber dynamics for the Ising model on complete multipartite graphs in the high temperature regime

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction and preliminaries

1.1. Notations

1.2. Ising model and Glauber dynamics

1.3. Markov chain mixing and cutoff phenomenon

1.4. Magnetization chain on complete multipartite graphs

1.5. Main results

Theorem 1.1 (Main result).

Theorem 1.2.

1.6. Organization of the article

2. Contraction of the magnetization chain in high temperatures

Proposition 2.1 (Magnetization chain).

Proof.

Remark.

Definition (Hamming distance).

Remark.

Lemma 2.2 (Contraction in mean for monotone coupling).

Proof.

Proposition 2.3.

Proof.

Lemma 2.4.

Proof.

Remark.

Lemma 2.5.

Proof.

Remark.

Lemma 2.6.

Proof.

Lemma 2.7.

Remark.

Proof.

Proposition 2.8.

Proof.

3. Variance bound of the magnetization in high temperatures

Lemma 3.1.

Proof.

Proposition 3.2.

Proof.

Lemma 3.3.

Proof.

Proposition 3.4 (Expected magnetization bound).

Proof.

4. Couplings

Definition (Modified matching).

Definition (Modified monotone update and coupling).

Remark.

Lemma 4.1 (Lemma 2.4, [10]).

Lemma 4.2 (Magnetization coupling).

Proof.

Definition (Good configurations).

Remark.

Lemma 4.3 (Lemma 3.3, [10]).

Definition (2​m2m-coordinate chain).

Lemma 4.4.

Proof.

Remark.

Lemma 4.5 (Post magnetization coupling).

Proof.

5. Upper and Lower Bounds in the high temperature regime

5.1. Upper Bound

Theorem 5.1.

Proof.

5.2. Lower Bound

Lemma 5.2 (Proposition 7.9, [11]).

Lemma 5.3.

Proof.

Theorem 5.4.

Proof.

6. Exponentially slow mixing in the low temperature regime

Lemma 6.1.

Proof.

Proof of \threflowtempresult.

Remark.

Acknowledgments.

References

Definition ( $2m$ -coordinate chain).