[Paper Review] Mixing times of critical 2D Potts models

Cannon, Sarah, Levin, David A.+1 more|arXiv (Cornell University)|Jul 7, 2016

Theoretical and Computational Physics44 references11 citations

TL;DR

This paper establishes the mixing time behavior of Glauber and Swendsen–Wang dynamics for the critical 2D q-state Potts model on the torus, showing polynomial inverse spectral gap for q = 3, quasi-polynomial for q = 4, and exponential for q > 4. It further proves that under free or monochromatic boundary conditions, Swendsen–Wang dynamics mixes in exp(no(1)) time for large q, contrasting with the torus where it is exponential, highlighting boundary condition sensitivity at criticality.

ABSTRACT

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.

Motivation & Objective

• To resolve the long-standing open problem of determining the mixing time of Glauber and Swendsen–Wang dynamics at the critical point βc(q) for the 2D Potts model.
• To clarify the role of boundary conditions in critical dynamics, particularly contrasting periodic, free, and monochromatic conditions.
• To establish sharp upper bounds on the inverse spectral gap for Glauber and Swendsen–Wang dynamics across the full range of q ∈ (1, ∞), distinguishing between continuous (q ≤ 4) and discontinuous (q > 4) phase transitions.
• To extend the RSW-type estimates and dynamical analysis from the Ising model (q=2) to q=3 and q=4, and to handle the breakdown of such estimates at q=4.

Proposed method

• Uses the Edwards–Sokal coupling to relate the Potts model to the random cluster (FK) model, enabling analysis via FK dynamics.
• Applies RSW-type estimates for FK models, extended to q=3 via recent results on crossing probabilities, and uses monotonicity and conditional independence to bound mixing times.
• Employs censored dynamics and block dynamics to decouple the system into manageable subregions (e.g., top and bottom halves), allowing recursive bounding of mixing time via coupling and total variation distance.
• Leverages self-duality of the FK model and comparison techniques to bound the distance to stationarity under different initial configurations and boundary conditions.
• Uses exponential moment bounds and union bounds on rare events (e.g., long dual crossings) to control error terms in coupling arguments.
• Applies the spectral gap inequality and mixing time bounds via the relationship between inverse spectral gap and L2-mixing time.

Experimental results

Research questions

• RQ1 What is the mixing time of Glauber dynamics for the critical 2D Potts model at q=3 on the torus?
• RQ2 How does the mixing time of Swendsen–Wang dynamics behave at q=4, where RSW estimates fail?
• RQ3 Does the critical slowdown for q>4 persist under free or monochromatic boundary conditions, and if so, how does it differ from the torus?
• RQ4 Can the dynamical behavior at criticality be faster under non-periodic boundary conditions, and if so, by how much?
• RQ5 How do the inverse spectral gaps of Glauber and Swendsen–Wang dynamics scale with system size n for q>4, and what does this imply about the nature of the phase transition?

Key findings

• For the 3-state Potts model on the torus, the inverse spectral gap of Glauber dynamics is at most nO(1), indicating polynomial mixing time.
• For the 4-state Potts model on the torus, the inverse spectral gap of Glauber dynamics is at most nO(log n), indicating quasi-polynomial mixing time.
• For every q>4 in the phase-coexistence regime, the inverse spectral gap of both Glauber and Swendsen–Wang dynamics on the torus is exponential in n, confirming critical slowdown.
• Under free or monochromatic boundary conditions and for large q>4, the inverse spectral gap of Swendsen–Wang dynamics is exp(no(1)), indicating sub-exponential mixing time.
• The results confirm that the critical slowdown for q>4 is sensitive to boundary conditions, with non-periodic conditions leading to significantly faster mixing than on the torus.
• The paper establishes matching upper bounds for Glauber dynamics on the FK model for all q∈(1,4], and provides a framework for extending dynamical analysis beyond the Ising model.