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Rapid phase ordering for Ising and Potts dynamics on random regular graphs

Reza Gheissari, Allan Sly, Youngtak Sohn

TL;DR

This work proves rapid phase ordering for low-temperature Ising and Potts Glauber dynamics on random $d$-regular graphs. It shows that a modest initial bias toward a dominant phase leads to fast quasi-equilibration to the corresponding metastable conditional measure in time $O(\log n)$, despite the exponential worst-case mixing in the regime where multiple phases compete. The authors develop a novel spacetime framework, introduce minus spacetime clusters and a rigid dynamics to bound their growth with exponential tails, and couple these results to the Potts dynamics via a dominating two-spin process. These techniques yield biased-initialization results that hold with high probability for $d\ge 7$ and $\beta$ above a $d$-dependent threshold, and they extend to the Potts case with similar bounds and a potentially vanishing bias as $d$ grows, highlighting a robust mechanism for fast phase ordering in ferromagnetic models on random graphs.

Abstract

We consider the Ising, and more generally, $q$-state Potts Glauber dynamics on random $d$-regular graphs on $n$ vertices at low temperatures $β\gtrsim \frac{\log d}{d}$. The mixing time is exponential in $n$ due to a bottleneck between $q$ dominant phases consisting of configurations in which the majority of vertices are in the same state. We prove that for any $d\ge 7$, from biased initializations with $ε_d n$ more vertices in state-$1$ than in other states, the Glauber dynamics quasi-equilibrates to the stationary distribution conditioned on having plurality in state-$1$ in optimal $O(\log n)$ time. Moreover, the requisite initial bias $ε_d$ can be taken to zero as $d \to \infty$. Even for the $q=2$ Ising case, where the states are naturally identified with $\pm 1$, proving such a result requires a new approach in order to control negative information spread in spacetime despite the model being in low temperature and exhibiting strong local correlations. For this purpose, we introduce a coupled non-Markovian rigid dynamics for which a delicate temporal recursion on probability mass functions of minus spacetime cluster sizes establishes their subcriticality.

Rapid phase ordering for Ising and Potts dynamics on random regular graphs

TL;DR

This work proves rapid phase ordering for low-temperature Ising and Potts Glauber dynamics on random -regular graphs. It shows that a modest initial bias toward a dominant phase leads to fast quasi-equilibration to the corresponding metastable conditional measure in time , despite the exponential worst-case mixing in the regime where multiple phases compete. The authors develop a novel spacetime framework, introduce minus spacetime clusters and a rigid dynamics to bound their growth with exponential tails, and couple these results to the Potts dynamics via a dominating two-spin process. These techniques yield biased-initialization results that hold with high probability for and above a -dependent threshold, and they extend to the Potts case with similar bounds and a potentially vanishing bias as grows, highlighting a robust mechanism for fast phase ordering in ferromagnetic models on random graphs.

Abstract

We consider the Ising, and more generally, -state Potts Glauber dynamics on random -regular graphs on vertices at low temperatures . The mixing time is exponential in due to a bottleneck between dominant phases consisting of configurations in which the majority of vertices are in the same state. We prove that for any , from biased initializations with more vertices in state- than in other states, the Glauber dynamics quasi-equilibrates to the stationary distribution conditioned on having plurality in state- in optimal time. Moreover, the requisite initial bias can be taken to zero as . Even for the Ising case, where the states are naturally identified with , proving such a result requires a new approach in order to control negative information spread in spacetime despite the model being in low temperature and exhibiting strong local correlations. For this purpose, we introduce a coupled non-Markovian rigid dynamics for which a delicate temporal recursion on probability mass functions of minus spacetime cluster sizes establishes their subcriticality.

Paper Structure

This paper contains 26 sections, 24 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The Potts Glauber dynamics case
  4. Proof ideas
  5. Evolution of the legacy region in spacetime
  6. Exponential tails on minus spacetime regions in the all-plus chain
  7. Extension to the Potts model
  8. Outline
  9. Confining negative information spread with plus initialization
  10. A general family of low-temperature two-spin dynamics
  11. Spacetime constructions and minus regions
  12. Internal measurability of minus spacetime clusters
  13. Trifurcation points and the rigid dynamics
  14. Exponential tails on minus spacetime clusters and regions
  15. Domination of conditional flip rates
  16. ...and 11 more sections

Key Result

Theorem 1

For every $d\ge 7$, there exists constants $C(\varepsilon, \beta, d)>0$, $\varepsilon_0(d)\in (0, 1)$ and $\beta_0<\infty$ with $\varepsilon_{0}(d) \asymp_d \frac{1}{\log d}$ and $\beta_0 \asymp_d \frac{\log d}{d}$, such that for every $\varepsilon\in (\varepsilon_0, 1]$ and every $\beta>\beta_0$, i By symmetry, we have the same bound on the distance to $\pi_G^- = \pi_G(\cdot \mid \Omega^-)$ if $X

Figures (2)

  • Figure 1: Spacetime diagrams for coupled Ising dynamics initialized from all-plus (left) and biased initialization $x_0$ (right). On the left, the minus spacetime clusters of $(X_t^+)_{t\ge 0}$ are highlighted orange; on the right the legacy spacetime cluster (union of minus clusters intersecting initial minus sites) is highlighted in magenta.
  • Figure 2: Snapshots of an Ising dynamics spacetime diagram initialized from biased initialization $x_0$ at times $t$ (left) and $t+\varepsilon$ (right) demonstrating the evolution of the legacy cluster $\mathcal{L}_{\le t}$ (highlighted purple). When the vertex $v$ adjacent to $\mathcal{L}_{t}$ flips from $+1$ to $-1$ at time $s\in [t,t+\varepsilon)$, the legacy cluster absorbs that vertex along with all minus spacetime clusters that are adjacent to $v$ at time $s$.

Theorems & Definitions (69)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Remark 1.2
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 59 more