A stochastic analysis approach to lattice Yang–Mills at strong coupling

We first recall the basic setup and definitions of the model.

Let $\Lambda_{L}={\mathbb{Z}}^{d}\cap L{\mathbb{T}}^{d}$ be a finite $d$ dimensional lattice with side length $L$ and unit lattice spacing, and we will consider various functions on it with periodic boundary conditions. We will sometimes write $\Lambda=\Lambda_{L}$ for short. We say that a lattice edge of ${\mathbb{Z}}^{d}$ is positively oriented if the beginning point is smaller in lexographic order than the ending point. Let $E^{+}$ (resp. $E^{-}$) be the set of positively (resp. negatively) oriented edges, and denote by $E_{\Lambda_{L}}^{+}$, $E_{\Lambda_{L}}^{-}$ the corresponding subsets of edges with both beginning and ending points in ${\Lambda_{L}}$. Define $E\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}E^{+}\cup E^{-}$ and let $u(e)$ and $v(e)$ denote the starting point and ending point of an edge $e\in E$, respectively.

We write $G$ for the Lie group $SO(N)$ or $SU(N)$ and $\mathfrak{g}$ for the associated Lie algebra $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$. Note that we always view $G$ as a real manifold (even for $SU(N)$), and $\mathfrak{g}$ as a real vector space, and we will write $d(\mathfrak{g})=\dim_{\mathbb{R}}\mathfrak{g}$.

To define the lattice Yang–Mills theory we need more notation, for which we closely follow [Cha] and [SSZloop].

A path is defined to be a sequence of edges $e_{1}e_{2}\cdots e_{n}$ with $e_{i}\in E$ and $v(e_{i})=u(e_{i+1})$ for $i=1,2,\cdots,n-1$. The path is called closed if $v(e_{n})=u(e_{1})$. A plaquette is a closed path of length four which traces out the boundary of a square. Also, let $\mathcal{P}_{\Lambda_{L}}$ be the set of plaquettes whose vertices are all in $\Lambda_{L}$, and $\mathcal{P}^{+}_{\Lambda_{L}}$ be the subset of plaquettes $p=e_{1}e_{2}e_{3}e_{4}$ such that the beginning point of $e_{1}$ is lexicographically the smallest among all the vertices in $p$ and the ending point of $e_{1}$ is the second smallest.

The lattice Yang-Mills theory (or lattice gauge theory) on ${\Lambda_{L}}$ for the structure group $G$, with $\beta\in\mathbb{R}$ the inverse coupling constant, is the probability measure $\mu_{\Lambda_{L},N,\beta}$ on the set of all collections $Q=(Q_{e})_{e\in E_{\Lambda_{L}}^{+}}$ of $G$-matrices, defined as

with {equ}[e:defS] S(Q) =defNβRe∑_p∈P^+_Λ_L Tr(Q_p), where $Z_{\Lambda_{L},N,\beta}$ is the normalizing constant, $Q_{p}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}Q_{e_{1}}Q_{e_{2}}Q_{e_{3}}Q_{e_{4}}$ for a plaquette $p=e_{1}e_{2}e_{3}e_{4}$, and $\sigma_{N}$ is the Haar measure on $G$. Note that for $p\in\mathcal{P}^{+}_{\Lambda_{L}}$ the edges $e_{3}$ and $e_{4}$ are negatively oriented, so throughout the paper we define $Q_{e}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}Q_{e^{-1}}^{-1}$ for $e\in E^{-}$, where $e^{-1}$ denotes the edge with orientation reversed. Also, ${\mathrm{Re}}$ is the real part, which can be omitted when $G=SO(N)$.

We will assume the following in our main results on lattice Yang–Mills.

Assumption 1.1 is equivalent to the following strong coupling assumption:

Define the (product) topological space ${\mathcal{Q}}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}G^{E^{+}}$, which will serve as our infinite volume configuration space. By Tychonoff’s theorem ${\mathcal{Q}}$ is compact. For each $a>1$ we define the distance $\rho_{\infty,a}$ on ${\mathcal{Q}}$ by {equ}[e:rho-inf] ρ_∞,a^2(Q,Q’)=def∑_e∈E^+1a—e—ρ^2(Q_e,Q_e’), with $|e|$ being the distance from $0$ to $e$ in ${\mathbb{Z}}^{d}$. Here the distances for different choices of $a$ give equivalent topologies, and we just write $\rho_{\infty}$ when there’s no confusion. ${\mathcal{Q}}$ is then a Polish space w.r.t. $\rho_{\infty}$. By standard results in topology, the topology induced by $\rho_{\infty}$ is equivalent with the product topology on ${\mathcal{Q}}$.

We can easily extend the measure $\mu_{\Lambda_{L},N,\beta}$ to the infinite volume configuration space ${\mathcal{Q}}$ by periodic extension, which is still denoted as $\mu_{\Lambda_{L},N,\beta}$. Namely, we can construct a random variable with law given by $\mu_{\Lambda_{L},N,\beta}$ and extend the random variable periodically, and the law of the periodic extension gives the desired extension of measure. Since $G$ and ${\mathcal{Q}}$ are compact, $\{\mu_{\Lambda_{L},N,\beta}\}_{L\geqslant 1}$ form a tight set.

We will consider the Langevin dynamic on ${\mathcal{Q}}$, formally given by {equ}[e:YM-formal] dQ = ∇S (Q) dt + 2dB , with $\mathfrak{B}=(\mathfrak{B}_{e})_{e\in E^{+}}$ being independent Brownian motions on $G$. This is formal since we will need to “extend” $\nabla{\mathcal{S}}$ to infinite volume in a suitable sense. More precisely, the Langevin dynamic we consider is the following SDE system parametrized by $e\in E^{+}$: {equs}[eq:YM in] dQ_e &= -12Nβ∑_p∈P,p≻e(Q_p-Q_p^*)Q_edt -12(N-1)Q_edt+2dB_eQ_e ,  if  G=SO(N) ,
dQ_e = -12Nβ∑_p∈P,p≻e( (Q_p-Q_p^*) - 1NTr(Q_p-Q_p^*) I_N) Q_edt
             -N2-1NQ_edt+2dB_eQ_e,     if  G=SU(N). Here $B=(B_{e})_{e\in E}$ is a collection of independent Brownian motions on the Lie algebra $\mathfrak{g}$ of $G$, and the terms linear in $Q_{e}$ arise from Casimir elements of the Lie algebras; we will review these in Section 2.

We will prove that there exists a unique probabilistically strong solutions to SDE (LABEL:eq:YM_in) starting from any initial data in ${\mathcal{Q}}$ in Proposition LABEL:lem:4.7. Hence the solutions form a Markov process in ${\mathcal{Q}}$ and the related semigroup is denoted by $(P_{t})_{t\geqslant 0}$.

Our first main result is as follows.

Here $W_{2}^{\rho_{\infty,a}}$ is the Wasserstein distance w.r.t. $\rho_{\infty,a}$ given for any $\mu,\nu\in{\mathscr{P}}({\mathcal{Q}})$

with ${\mathscr{C}}(\mu,\nu)$ being the set of couplings between $\mu$ and $\nu$. Remark that $\widetilde{K}_{\mathcal{S}}$ can be explicitly given by (LABEL:e:tildeK) below and gives a lower bound of spectral gap for $(P_{t})_{t\geqslant 0}$ in Wasserstein distance. In Theorem 1.4 we will see that $K_{\mathcal{S}}$ gives a lower bound of spectral gap in $L^{2}(\mu^{\textnormal{\tiny{ym}}}_{N,\beta})$.

We remark that uniqueness for small $\beta$ could possibly also be proven using the method of Dobrushin, see e.g. [Dobrushin1970]. To this end one would also need to consider the related Wasserstein metric with respect to the Riemannian distance similarly as we do in this paper. However as we understand such an argument has not been carried out in detail for lattice Yang Mills in the literature. Here we give a proof based on a new idea which is a variant of Kendall–Cranston’s coupling used earlier in the stochastic analysis on manifold.

The idea for the proof of Theorem 1.2 is to use finite dimensional approximation, for which we construct a suitable coupling and find a suitable distance such that the associated Wasserstein distance between the two finite dimensional approximations starting from different initial distributions decays exponentially fast in time with uniform speed.

We define the cylinder functions $C^{\infty}_{cyl}({\mathcal{Q}})$ by {equ}[e:cylQ] C^∞_cyl(Q) ={ F: F=f(Q_e_1,…, Q_e_n),n∈N, e_i∈E^+, f∈C^∞(G^n)}. We then obtain the following log-Sobolev inequality for $\mu^{\textnormal{\tiny{ym}}}_{N,\beta}$ based on Bakry–Émery’s criterion.

Theorem 1.4 follows from Theorem 1.2 and Corollary LABEL:co:lo, which states the log-Sobolev inequality for every tight limit of $(\mu_{\Lambda_{L},N,\beta})_{L\geqslant 1}$. The RHS of (1.3) is the Dirichlet form associated with the Langevin dynamic (see Proposition LABEL:th:D). Hence, (1.3) holds for the functions in the domain of Dirichlet form by lower-semicontinuity.

As some simple applications of the Poincaré inequality, we show certain “susceptibility” bounds on the field $Q_{e}$ and ${\rm Tr}(Q_{p})$, see Corollaries LABEL:co:c and LABEL:co:cc. These examples demonstrate how to choose suitable functions in these functional inequalities to yield interesting bounds for the model.

Log-Sobolev and Poincaré inequalities in Theorem 1.4 follow by checking the Bakry–Émery criteria [MR889476, BakryLedoux] directly for the finite dimensional approximation on the product manifolds. As the Ricci curvatures of the target manifolds $G$ are given by positive constants, so are the Ricci curvatures of the configuration space (i.e. the product manifolds). The Hessian of the Hamiltonian could also be bounded by the Ricci curvatures in the strong coupling regimes.

As a corollary of Theorem 1.4 we obtain the following large $N$ properties of the Wilson loops. For the rest of this paper, by a loop we mean an equivalent class of closed paths (as defined in Section 1.1), where the equivalence relation $\sim$ is given by cyclic permutations $e_{1}e_{2}\cdots e_{n}\sim e_{i}e_{i+1}\cdots e_{n}e_{1}e_{2}\cdots e_{i-1}$ for any $i\in\{1,...,n\}$, and it has no two successive edges of the form $e^{-1}e$. We will always assume that a loop is non-empty, i.e. has positive number of edges. Given a loop $\ell=e_{1}e_{2}\cdots e_{n}$, recall that the Wilson loop variable $W_{\ell}$ is defined as {equ}[e:loop] W_ℓ=defTr(Q_e_1Q_e_2⋯Q_e_n) .

Corollary 1.5 is proven in Section LABEL:sec:3.2. Our proof is novel which is based on the Poincaré inequality (LABEL:e:PoinYM). Note that our formulation of the result is different from [Cha] and [Jafar] in which the factorization property of Wilson loops was obtained by taking a sequence of increasing finite lattices ${\mathbb{Z}}^{d}=\cup_{N=1}^{\infty}\Lambda_{N}$, considering the correlations of Wilson loops over $\Lambda_{N}$, and taking infinite volume limit simultaneously as the large $N$ limit when sending $N\to\infty$. In our approach, we work directly in infinite volume, which seems more natural. The subtlety here, as mentioned above and also explained in [Cha], is that the ’t Hooft coupling places $N\beta$ instead of $\beta$ in front of the Hamiltonian so one would require $N\beta$ to be sufficiently small to obtain the infinite volume limit, which would appear to be problematic when taking the large $N$ limit afterwards. However, thanks to our precise smallness condition on $\beta$ in (1.2), we can take $\beta$ small uniformly in $N$. This also allows us to derive bounds on the variances of Wilson loops which are explicit in terms of $N$. Our proof based on the Poincaré inequality which follows from the Bakry–Émery condition also appears to be simpler than the arguments in aforementioned previous work.

Furthermore, we obtain the following exponential decay property of the covariance. Consider $f\in C^{\infty}_{cyl}({\mathcal{Q}})$ and we write $\Lambda_{f}$ for the set of the edges $f$ depends on. Let $|\Lambda_{f}|$ denote the cardinality of $\Lambda_{f}$. We define

where $C^{\infty}_{cyl}({\mathcal{Q}})$ is introduced in (LABEL:e:cylQ) and $\nabla_{e}$ is introduced in Section LABEL:sec:3.1. We also write $d(A,B)$ for the distance between $A,B\subset E^{+}$, which is given by the nearest distance between the vertices in $A$ and $B$.

Note that $f$ and $g$ in the above corollary can be chosen to be Wilson loops, or functions of an arbitrary number of Wilson loops, which are of particular interest in physics. We also remark that exponential decay of correlations (together with certain center symmetry conditions) is also related to Wilson’s area law for Wilson loops (see [MR4278289, Theorem 2.4]). The proof of Corollary 1.6 is given in Section LABEL:sec:3.2.

We conclude this subsection by some brief comments on the challenges or subtleties in the proofs of the above results. One of the important ingredients in the proofs is to estimate the Hessian or the general second order derivatives for the interaction ${\mathcal{S}}$ defined in (LABEL:e:defS). Note that a term of the form $N{\rm Tr}(Q_{p})$ in the interaction ${\mathcal{S}}$ would “appear” to be of order $N^{2}$, which would be too large for us to obtain the desired results. In our proofs we will properly arrange terms and apply certain properties of the Lie groups and we will show that the relevant second order derivatives are actually at most of order $|\beta|N$. See the explanations before Lemma LABEL:lem:4.1 and the proof of Theorem 1.2 in Section LABEL:sec:uniqueYM for more details. This is one of the crucial reasons which allow us to compare the Ricci curvatures of the Lie groups and Hessians to verify the Bakry–Émery condition by choosing $\beta$ small, and also prove ergodicity using a suitable weighted distance.

The study of properties of lattice gauge theories recently attracts much interest. Besides the aforementioned work by [Cha] and [Jafar], [Chatterjee16] computed the leading terms of free energies, [MR3861073] provided an elaborate description of loop expectations in the planar setting, and [ChatterjeeJafar] derived $1/N$ expansions in the $SO(N)$ case at strong coupling. Wilson loops (and also Wilson lines when coupled with Higgs) for gauge theories with finite structure groups were studied in [MR4107931, FLV20, FLV21, Cao20, forsstromAbelian, Adhikari2021]; see also [GS21] for the $U(1)$ case. Moreover, exponential correlation decay for lattice gauge theories with finite abelian structure groups was obtained by [Forsstrom2021] using coupling argument, and for finite non-abelian structure groups this was proved by [DdhikariCao].

Our present article provides a new approach to study these models via stochastic analysis and dynamical perspective; see also [SSZloop] for a new derivation of loop i.e. Makeenko–Migdal equations for Wilson loops by such methods.

We remark that the choice of constant positive curvature Lie groups $SO(N)$ and $SU(N)$ in this article is a technical simplification for demonstrating our method, and it should apply as well for other compact target spaces with constant or non-constant positive curvatures. For instance it should apply to a lattice $SO(N)$ Yang–Mills model coupled with a Higgs field $\Phi$ which takes values in a sphere in $\mathbb{R}^{N}$ (i.e. rotator model) via a gauge-covariant derivative term, whose action takes the form ${\mathrm{Re}}\sum_{p}\mathrm{Tr}(Q_{p})+\sum_{e}|Q_{e}\Phi_{v(e)}-\Phi_{u(e)}|^{2}$.

It would certainly be interesting to show if log-Sobolev inequalities still hold when the lattice spacing vanishes, in the situations where the continuum limits of these models are shown or expected to exist. In this direction, on the two dimensional torus, the continuum limit of lattice approximations of the Yang–Mills measures on 1-forms was recently obtained by Chevyrev [Chevyrev19YM], who also showed that a certain class of Wilson loop observables of this random 1-form coincide in law with the corresponding observables under the Yang–Mills measure in the sense of [Levy03]. Note that the Langevin dynamics for Yang–Mills models on the two and three dimensional continuous torus were recently constructed in [CCHS2d, CCHS3d] (see [ChevyrevReview] for a review of these results), and as mentioned in [CCHS2d] it would be interesting to show that the lattice dynamics of the type (LABEL:e:YM-formal) converge to the processes constructed in the above papers in two and three dimensions. For some of the recent progress along this direction, see the proof of log-Sobolev inequalities for the $\Phi^{4}_{2,3}$ and sine-Gordon models [RolandPhi4, RolandSG], and the 1D nonlinear $\sigma$-model (see [AnderssonDriver, Hairer16, StringManifold]) for which the log-Sobolev inequality, ergodicity and non-ergodicity (depending on the curvature of the target manifolds) were obtained in [RWZZ17, CWZZ18].

Notation. Given a Polish space $E$, we write $C([0,T];E)$ for the space of continuous functions from $[0,T]$ to $E$. We use ${\mathscr{P}}(E)$ to denote all the probability measures on $E$ with Borel $\sigma$-algebra.