Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fixed-magnetization Ising on random graphs up to reconstruction

Reza Gheissari, Will Perkins, Corrine Yap

Abstract

We study the fixed-magnetization ferromagnetic Ising model on random $d$-regular graphs for $d\ge 3$ and inverse temperature below the tree reconstruction threshold. Our main result is that for each magnetization $η$, the free energy density of the fixed-magnetization Ising model converges to the annealed free energy density, itself the Bethe free energy of an Ising measure on the infinite $d$-regular tree. Moreover, the fixed-magnetization Ising model exhibits local weak convergence to this tree measure. A key challenge to proving these results is that for magnetizations between the model's spinodal points, the limiting tree measure corresponds to an unstable fixed point of the belief propagation equations. As an application, we prove that the positive-temperature Zdeborová--Boettcher conjecture on max-cut and min-bisection holds up to the reconstruction threshold: on the random $d$-regular graph, the expected fraction of bichromatic edges in the anti-ferromagnetic Ising model plus the expected fraction of bichromatic edges in the zero-magnetization ferromagnetic Ising model equals $1+o(1)$. A second application is completely determining the large deviation rate function for the magnetization in the Ising model on the random regular graph up to reconstruction. Finally, we use the precise understanding of this rate function to show that the Glauber dynamics for the full Ising model on the random graph mixes in sub-exponential time from uniformly random initialization, well into the non-uniqueness regime where the worst-case initialization mixing time is exponentially slow.

Fixed-magnetization Ising on random graphs up to reconstruction

Abstract

We study the fixed-magnetization ferromagnetic Ising model on random -regular graphs for and inverse temperature below the tree reconstruction threshold. Our main result is that for each magnetization , the free energy density of the fixed-magnetization Ising model converges to the annealed free energy density, itself the Bethe free energy of an Ising measure on the infinite -regular tree. Moreover, the fixed-magnetization Ising model exhibits local weak convergence to this tree measure. A key challenge to proving these results is that for magnetizations between the model's spinodal points, the limiting tree measure corresponds to an unstable fixed point of the belief propagation equations. As an application, we prove that the positive-temperature Zdeborová--Boettcher conjecture on max-cut and min-bisection holds up to the reconstruction threshold: on the random -regular graph, the expected fraction of bichromatic edges in the anti-ferromagnetic Ising model plus the expected fraction of bichromatic edges in the zero-magnetization ferromagnetic Ising model equals . A second application is completely determining the large deviation rate function for the magnetization in the Ising model on the random regular graph up to reconstruction. Finally, we use the precise understanding of this rate function to show that the Glauber dynamics for the full Ising model on the random graph mixes in sub-exponential time from uniformly random initialization, well into the non-uniqueness regime where the worst-case initialization mixing time is exponentially slow.

Paper Structure

This paper contains 28 sections, 39 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free energy and local weak limit of fixed magnetization Ising
  3. Ising Glauber dynamics from a uniformly random initialization
  4. Proof outline
  5. Free energy density: background
  6. Free energy density for unstable fixed points
  7. Local weak convergence
  8. Analysis of Ising Glauber dynamics
  9. Ising model on trees and first-moment calculations
  10. Ising model on trees and BP fixed points
  11. Reconstruction threshold
  12. Annealed free energy density
  13. Free energy of the fixed-magnetization Ising model
  14. Truncated second moment
  15. The planted model
  16. ...and 13 more sections

Key Result

Theorem 1.1

Fix $d \ge 3$ and suppose $0 \le \beta < \beta_r$. Then with high probability over $G \sim {\bf G}_d(n)$ it holds for every $\eta \in [-1,1]$ that

Figures (2)

  • Figure 1: For $d=10$, $\beta=.32 \in (\beta_c,\beta_r)$, the plot of $f_{d,\beta}(\eta)$ as a function of $\eta$. The typical magnetization is $\eta_*$ and the spinodal point $\eta_s$ is the smallest magnetization achievable by a stable BP fixed point for some external field.
  • Figure 2: For $d=10$, $\beta=.32$, $f_{d,\beta}(\eta) + \eta h$ is plotted for different values of $\eta$ and $h$. This is the contribution to the annealed free energy of the Ising model with external field from different values of $\eta$. $\eta$ ranges from $-1$ to $1$ along the x-axis, while the different curves are for values of $h$ ranging from $0$ (blue) to $.2$ (red). Each value of $\eta$ appears as a stationary point for a unique value of $h$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1: georgii2011gibbs
  • proof : Proof of Lemma \ref{['lem:existenceTreeMeasure']}
  • Lemma 2.2
  • ...and 65 more