Failure of Ornstein–Zernike asymptotics for the pair correlation function at high temperature and small density

An anisotropic version of the OZ asymptotics (2) has been rigorously derived for finite-range ferromagnetic Ising models on $\mathbb{Z}^{d}$, both above the critical temperature [5, 6] and when the magnetic field is non-zero [7]: in both cases, as $\left\|x\right\|$ tends to infinity,

where both $A_{\beta,h}$ and the inverse correlation length $\nu_{\beta,h}$ are positive, analytic functions of the direction $\hat{x}=x/\left\|x\right\|$. The same behavior still holds as long as the interaction decays super-exponentially [8]. Of course, this cannot be true for interactions decaying sub-exponentially, since the pair correlation function cannot decay slower than the interaction, as seems to have been first noted by Widom [9].

Let us thus restrict our attention to coupling constants $J_{x,y}=J_{y-x}$ of the form

for some $\alpha\in\mathbb{R}$ and $c>0$. (More general forms will be considered below.) Let us also assume for simplicity that there is no magnetic field (and omit it from the notation). We emphasize, however, that the analysis below can also be done by fixing some arbitrary $\beta$ and looking at the behavior of $h\mapsto\nu_{\beta,h}(s)$.

Let $s$ be a unit vector in $\mathbb{R}^{d}$ and denote the inverse correlation length in direction $s$ by

where $[ns]$ is a vertex of $\mathbb{Z}^{d}$ which is closest to $ns\in\mathbb{R}^{d}$. It is known [10] that $\nu_{\beta}(s)>0$ for all $\beta<\beta_{\rm c}$, where $\beta_{\rm c}$ denotes the inverse critical temperature. In addition, it is easy to prove that $\nu_{\beta}(s)$ is decreasing in $\beta$ and satisfies

Moreover, $\lim_{\beta\uparrow\beta_{\rm c}}\nu_{\beta}(s)=0$ [11]. The behavior in (5) is in sharp contrast with what happens when the interaction decays super-exponentially (in which case $\lim_{\beta\to 0}\nu_{\beta}(s)=\infty$) and opens up the possibility that $\nu_{\beta}(s)$ saturates at a finite temperature: $\nu_{\beta}(s)=c\left\|s\right\|=c$ for all $\beta<\beta_{\rm sat}(s)$, for some $\beta_{\rm sat}(s)\in(0,\beta_{\rm c})$.

Observe that, if the latter were to occur, then there would be several remarkable consequences:

• •

  The inverse correlation length $\nu_{\beta}(s)$ would not depend analytically on the temperature in the whole high-temperature regime $(0,\beta_{\rm c})$.

• •

  One of the basic assumptions of OZ theory, namely that the rate of decay of the pair correlation function is strictly slower than the rate of decay of the interaction (in the present case, $\nu_{\beta}(s)<c\left\|s\right\|$) would be violated.

It turns out that such a saturation phenomenon can occur, in any dimension. Whether or not it does so depends on the value of the exponent $\alpha$. Namely, saturation occurs (that is, $\beta_{\rm sat}(s)>0$) if and only if

In particular, even when $d=1$, the inverse correlation length is a non-analytic function of $\beta$ whenever $\alpha<-1$, which contrasts with the well-known analyticity of all thermodynamic quantities [12, 13].

As mentioned above, in the whole saturation regime $\beta<\beta_{\rm sat}(s)$, one of the basic assumptions of the OZ theory fails to hold. In fact, one can show [8] that, while the OZ asymptotic behavior (3) still holds whenever $\beta\in(\beta_{\rm sat}(s),\beta_{\rm c})$, it fails when $\beta<\beta_{\rm sat}(s)$, where the pair correlation decays proportionally to the coupling constant.

Let us now consider coupling constants of the form

where $\lvert\cdot\rvert$ is an arbitrary norm on $\mathbb{R}^{d}$ and the prefactor $\psi$ is positive and sub-exponential, in the sense that $\lim_{\lvert x\rvert\to\infty}\ln(\psi(x))/\lvert x\rvert=0$. We also assume that it behaves in an approximately isotropic manner: there exists $c_{1}>0$ and a function $\psi_{0}:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ such that $\frac{1}{c}_{1}\psi_{0}(\left\|x\right\|)\leq\psi(x)\leq c_{1}\psi_{0}(\left\|x\right\|)$ for all $x\neq 0$.

In order to ease the formulation of the extension of (6) to this more general framework, we need to make an additional genericity assumption on the direction $s$. Let $\mathscr{U}$ be the unit-ball associated to the norm $\lvert\cdot\rvert$. Given a direction $s$, consider the corresponding point $s_{0}=s/\lvert s\rvert\in\partial\mathscr{U}$. By convexity, every $u\in\partial\mathscr{U}$ close enough to $s_{0}$ can be written as

where $\tau>0$ is some small real number, $v$ is unit vector in the tangent plane to $\partial\mathscr{U}$ at $s_{0}$, $\hat{t}$ is a unit vector normal to the tangent plane and $f$ is some non-negative convex function defined on the tangent plane such that $f(0)=0$ (see Figure 1).

Our assumption on the direction $s$ is that the qualitative behavior of $f$ is the same in all directions $v$: there exist $c_{2}>0$ and a non-decreasing convex function $g$ such that, for any unit vector $v$ in the tangent plane and $\tau$ small enough, we have

When this holds, we will say that $\partial\mathsf{U}$ is quasi-isotropic in direction $s$. Examples when quasi-isotropy is satisfied in every direction $s$ include $p$-norms with $p\in[1,+\infty]$.

Our main result on whether $\beta_{\rm sat}(s)>0$ is the following:

(In (10), we make the convention $g^{-1}\equiv 1$ when $g\equiv 0$.)

When considering (4), we can take $\psi_{0}(x)=\left\|x\right\|^{\alpha}$ and $g(\tau)=\tau^{2}$; it is thus clear that (10) reduces to (6) in that case.

In this section, we illustrate on a specific example the direction-dependence of the saturation phenomenon.

Let us consider the model on $\mathbb{Z}^{2}$ with interaction

where $\left\|\cdot\right\|_{p}$ is the $p$-norm and we assume that $p\in(2,\infty)$.

Let $s$ be a unit vector in $\mathbb{R}^{2}$ and let us write $s_{0}=(x_{0},y_{0})=s/\left\|s\right\|_{p}$. When both $x_{0}$ and $y_{0}$ are nonzero, the local parametrization (8) of $\partial\mathscr{U}$ at $s_{0}$ applies with

so that one can choose $g(\tau)=\tau^{2}$. Condition (10) then implies that there is saturation in direction $s$ if and only if $\alpha<-3/2$, as in (6) (which should not be surprising, since $\partial\mathscr{U}$ has the same qualitative behavior in both cases).

In the remaining cases ($s=\pm(1,0)$ or $\pm(0,1)$), the local parametrization (8) of $\partial\mathscr{U}$ at $s_{0}$ applies with

so that one can choose $g(\tau)=\tau^{p}$. Therefore, in this case, there is saturation in direction $s$ if and only if

This clearly shows that the occurrence of saturation depends on both the temperature and the direction. In particular, when $-3/2>\alpha\geq\frac{1}{p}-2$, $\beta_{\rm sat}(s)=0$ when $s\in\{\pm(1,0),\pm(0,1)\}$ but is positive for all other directions.

When $\beta<\beta_{\rm c}$, the 2-point function of the Ising model admits a graphical representation as a sum over paths:

where the sum is over edge-self-avoiding paths connecting $0$ to $x$ and $\mathsf{q}_{\beta}(\gamma)$ is a suitable non-negative weight [14]. Suppose that $\beta_{\rm sat}(\hat{x})>0$. It turns out that the saturation phenomenon occurring at $\beta_{\rm sat}(\hat{x})$ corresponds to a change of behavior of the typical paths contributing to the sum in (11). When $\beta\in(\beta_{\rm sat}(\hat{x}),\beta_{\rm c})$, typical paths $\gamma$ have a length of order $\left\|x\right\|$ and behave diffusively, whereas when $\beta\in(0,\beta_{\rm sat}(\hat{x}))$, typical paths $\gamma$ contain $\mathsf{O}(1)$ steps: there is a giant step connecting a vertex close to $0$ to a vertex close to $x$, see Fig. 2. This change of behavior is reminiscent of condensation phenomena for sums of random variables; we refer to [15] for a review.

In this section, we prove Theorem 1 when $d=1$. The proof of the full Theorem 1 can be found in [4].

We consider the Ising model on $\mathbb{Z}$ with interactions as in (7); for simplicity, we assume that $\psi(y)=\psi(|y|)$ for all $y$ and that $\psi$ is monotone in $\mathbb{R}_{>0}$. Introduce the generating functions (Laplace transforms)

Note that $\mathbb{G}$ has radius of convergence $\nu_{\beta}(1)$, while $\mathbb{J}$ has radius of convergence $|1|$. Moreover, since we are assuming that $d=1$, condition (10) is equivalent to $\sum_{y\in\mathbb{Z}}\psi(y)<\infty$ and to $\mathbb{J}(|1|)<\infty$.

Our analysis below will be based on the following inequalities satisfied by the weights in (11) (see [14]): there exists $C_{\beta}>0$ such that, if $\gamma=(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$,

Note that the lower bound immediately implies that $\nu_{\beta}(\hat{x})\leq\lvert\hat{x}\rvert$ (simply consider the path $\gamma$ composed of a single step from $0$ to $x$ in (11)).

In this section, we illustrate, in a simple example, the mechanism leading to a violation of OZ asymptotics when saturation occurs. Let us thus consider the Ising model on $\mathbb{Z}$ with coupling constants of the form

(Again, we refer to [4] for a treatment of the general case in any dimension.)

In the previous section, it was shown that saturation occurs when

We are going to show that there exists $c_{\beta}>0$ such that

The lower bound follows directly from (11) by considering the path composed of a single step from $0$ to $x$. For the upper bound, note first that (12) implies

where in the last line, we assumed that $y_{1}$ is the vector with maximal $\lvert\cdot\rvert$-norm and used that $\lvert y_{1}\rvert+\dots+\lvert y_{n}\rvert\geq\lvert y_{1}+\dots+y_{n}\rvert=\lvert x\rvert$. Observe now that, for any two real numbers $a,b$ with $a\geq b$, we have

since $\alpha$ is negative (by (13)). Using this observation $n-1$ times, it follows that

where we used $\lvert y_{1}\rvert\geq\max_{i}\lvert y_{i}\rvert$ for the first inequality and $\lvert y_{1}\rvert+\dots+\lvert y_{n}\rvert\geq\lvert x\rvert$ and $\alpha<0$ for the second one.

Using (16) in (15) yields

Thanks to Condition (13), we conclude that the sum over $n$ is convergent when $\beta$ is small enough, which yields the upper bound in (14).

We have considered an Ising model with exponentially decaying interactions and have explained how the latter can, in suitable circumstances, undergo a saturation “transition”, in which the rate of exponential decay of the pair correlation function coincides with the rate of exponential decay of the interaction over a non-empty interval of temperatures $(0,\beta_{\rm sat})$ (where $\beta_{\rm sat}\in(0,\beta_{\rm c})$ is in general direction-dependent). Since the inverse correlation length becomes independent of the temperature below $\beta_{\rm sat}$, it is not an analytic function of $\beta$ on $(0,\beta_{\rm c})$. When this occurs, the pair correlation function fails to exhibit Ornstein–Zernike asymptotics for all $\beta<\beta_{\rm sat}$.

A number of issues remain open. We have limited our analysis, for technical reasons, to ferromagnetic systems. It is unclear to us whether the general phenomenology changes when this assumption is dropped. In addition, the behavior of the system at $\beta_{\rm sat}$ is not understood, although it can be shown that in one-dimensional ferromagnetic systems, saturation does not occur at $\beta_{\rm sat}$ [8].