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A spectral condition for spectral gap: fast mixing in high-temperature Ising models

Abstract

We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than 1. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models.

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Acknowledgements

We would like to thank Fanny Augeri for enlightening discussions. We also thank Roland Bauerschmidt for some useful comments. We thank Ahmed El Alaoui, Heng Guo, Vishesh Jain, and the anonymous reviewers for useful feedback.

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Correspondence to Frederic Koehler.

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R. Eldan: Supported by a European Research Council Starting Grant (ERC StG) Grant Agreement No. 803084 and by an Israel Science Foundation Grant No. 715/16. F. Koehler: This work was supported in part by NSF CAREER Award CCF-1453261, NSF Large CCF-1565235, Ankur Moitra’s ONR Young Investigator Award, and European Research Council (Grant No. 803084). O. Zeitouni: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 692452)

Inequivalence of Dirichlet forms

Inequivalence of Dirichlet forms

The Dirichlet form \({\mathcal {E}}_{\nu }(\varphi ,\varphi )\) can be viewed as the expected norm squared for an appropriate notion of gradient of \(\varphi \). On the other hand, another natural notion of discrete gradient for functions on the hypercube is given by \((\nabla \varphi )_i(x) = \varphi (x_{\sim i}, x_i = 1) - \varphi (x_{\sim i}, x_i = -1)\) which is used in the result of Bauerschmidt and Bodineau [3], and the norm of this discrete gradient squared is the Dirichlet form of a different semigroup with a variable transition rate [17]. In the case of the SK model, the transition rate of the variable-rate chain is sometimes exponentially large in \(\sqrt{n}\); in what follows, we give a simple example of a function \(\varphi \) witnessing that the Dirichlet forms similarly can differ in size by an exponentially large factor in \(\sqrt{n}\).

To compare these two Dirichlet forms, we have the following estimates which follow immediately from (27):

$$\begin{aligned} \left( \min _{x \sim y} \frac{\nu (y)}{\nu (x) + \nu (y)}\right) {\mathbb {E}}_{\nu } \Vert \nabla \varphi \Vert ^2 \lesssim {\mathcal {E}}_{\nu }(\varphi ,\varphi ) \lesssim {\mathbb {E}}_{\nu } \Vert \nabla \varphi \Vert ^2. \end{aligned}$$

In the context of the SK Model, the parenthesized term is of size \(e^{-\Theta (\beta \sqrt{n})}\) and both estimates are tight up to constants. To see this for the lower bound, define \(a \in \{\pm 1\}^n\) by \(a_1 = -1\) and \(a_j = \text {sgn}(J_{1j})\) otherwise; the significance of this choice is that in the SK model, it’s exponentially unlikely to see \(X_1 = a_1\) given \(X_{\sim 1} = a_{\sim 1}\). Let \(\lambda \) be an atomic measure supported on a, so

$$\begin{aligned} \frac{d\lambda }{d\nu }(x) = \frac{{\mathbf {1}}[x = a]}{\nu (a)}. \end{aligned}$$

If we define \(\varphi = \sqrt{\frac{d\lambda }{d\nu }}\) then, see (28),

$$\begin{aligned} {\mathcal {E}}\left( \varphi ,\varphi \right) = \sum _{x \sim y} \frac{\nu (x)\nu (y)}{\nu (x) + \nu (y)} (\varphi (x) - \varphi (y))^2 = \sum _{y : y \sim a} \frac{\nu (y)}{\nu (a) + \nu (y)} \le n. \end{aligned}$$

In comparison, for the discrete gradient \(\nabla \varphi \) we have

$$\begin{aligned} {\mathbb {E}}|\nabla \varphi (X)|^2 \ge \frac{e^{c\beta \sqrt{n}} \nu (a)}{\nu (a)} = e^{\Theta (\beta \sqrt{n})} \end{aligned}$$

where the lower bound follows by considering the \(a'\) which equals a but flipped on the first coordinate, and which (from the definition of the SK model) is \(e^{\Theta (\beta \sqrt{n})}\) more likely under \(\nu \).

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Eldan, R., Koehler, F. & Zeitouni, O. A spectral condition for spectral gap: fast mixing in high-temperature Ising models. Probab. Theory Relat. Fields 182, 1035–1051 (2022). https://doi.org/10.1007/s00440-021-01085-x

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Mathematics Subject Classification

  • 60J10
  • 60J27