[Paper Review] Critical Ising on the square lattice mixes in polynomial time

Eyal Lubetzky, Allan Sly|arXiv (Cornell University)|Jan 11, 2010

Stochastic processes and statistical mechanics43 references4 citations

TL;DR

This paper establishes the first rigorous polynomial upper bound for the spectral gap of the Glauber dynamics in the critical two-dimensional Ising model on the square lattice, confirming the conjectured critical slowdown. By leveraging recent advances in the scaling limit of the critical Fortuin-Kasteleyn representation and Markov chain analysis, it proves the inverse-gap is polynomial in the side-length, independent of boundary conditions.

ABSTRACT

The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical $β_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A seminal series of papers verified this on $\Z^2$ except at $β=β_c$ where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $β=β_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

Motivation & Objective

• To resolve the longstanding open problem of the critical slowdown in the Glauber dynamics of the 2D Ising model at inverse-temperature βc.
• To establish a polynomial upper bound on the inverse-spectral gap at criticality, confirming the conjectured critical slowdown behavior.
• To extend the result to rectangles with unbounded aspect ratio and to periodic and anti-ferromagnetic boundary conditions.
• To provide a rigorous connection between static critical geometry (via SLE and CLE) and dynamic mixing properties of the Ising model.

Proposed method

• Utilizes the critical Fortuin-Kasteleyn (FK) random-cluster representation of the Ising model, exploiting its conformal invariance and scaling limit properties.
• Applies spatial mixing techniques based on the probability of crossing events in the FK model, bounding the dependence on boundary conditions.
• Employs block dynamics and recursive analysis to reduce the problem to smaller domains with fixed boundary conditions.
• Uses coupling arguments to show that with positive probability, configurations in adjacent blocks can be coupled in one step, ensuring bounded inverse-gap for block dynamics.
• Applies the spectral gap relation between single-site and block dynamics to transfer the bound to the original Glauber dynamics.
• Extends results to periodic boundary conditions by splitting the torus into overlapping blocks with one pair of periodic boundaries and using spatial mixing under such conditions.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the spectral gap of the Glauber dynamics for the critical Ising model on Z²?
• RQ2 Does the inverse-spectral gap grow polynomially in the side-length at criticality, as conjectured?
• RQ3 Can the critical slowdown be rigorously confirmed using modern tools from conformally invariant systems and SLE?
• RQ4 How does the mixing time depend on boundary conditions, especially for periodic or anti-ferromagnetic settings?
• RQ5 Can the polynomial bound be extended to rectangles with unbounded aspect ratio?

Key findings

• The inverse-spectral gap of the Glauber dynamics for the critical Ising model on a finite box of side-length n is bounded above by n^C for some absolute constant C > 0, independent of boundary conditions.
• The total-variation mixing time of the Glauber dynamics is bounded above by a polynomial in n, confirming rapid mixing at criticality.
• For rectangles with unbounded aspect ratio, the inverse-gap is bounded by a polynomial in the shorter side-length only.
• The inverse-gap is shown to grow at least as fast as n^{7/4}, providing a lower bound consistent with the conjectured universality of the exponent.
• The result extends to the critical anti-ferromagnetic Ising model under arbitrary boundary conditions, due to symmetry under spin-flip transformation.
• The proof establishes a uniform bound on the inverse-gap for tori with periodic boundary conditions by decomposing the system into overlapping blocks with one pair of periodic boundaries.