[Paper Review] Optimal Mixing Time for the Ising Model in the Uniqueness Regime

Xiaoyu Chen, Weiming Feng+2 more|arXiv (Cornell University)|Nov 4, 2021

Markov Chains and Monte Carlo Methods25 references4 citations

TL;DR

This paper establishes an optimal $O(n\log n)$ mixing time for the Glauber dynamics in the uniqueness regime of the Ising model on general graphs, even with unbounded maximum degree $\Delta$. By refining a boosting technique using entropic independence and a novel modified log-Sobolev (MLS) constant theorem, the authors achieve a tight bound that matches the theoretical lower bound, resolving a long-standing open problem in statistical physics and MCMC sampling.

ABSTRACT

We prove an optimal $O(n \log n)$ mixing time of the Glauber dynamics for the Ising models with edge activity $β\in \left(\frac{Δ-2}Δ, \fracΔ{Δ-2} ight)$. This mixing time bound holds even if the maximum degree $Δ$ is unbounded. We refine the boosting technique developed in [CFYZ21], and prove a new boosting theorem by utilizing the entropic independence defined in [AJK+21]. The theorem relates the modified log-Sobolev (MLS) constant of the Glauber dynamics for a near-critical Ising model to that for an Ising model in a sub-critical regime.

Motivation & Objective

• Address the open problem of closing the gap between known polynomial mixing time bounds and the theoretical $\Omega(n\log n)$ lower bound for Ising model Glauber dynamics in the uniqueness regime.
• Establish an optimal $O(n\log n)$ mixing time bound that holds for both ferromagnetic and anti-ferromagnetic Ising models on general graphs with unbounded maximum degree $\Delta$.
• Refine the boosting technique from prior work to achieve tighter bounds by leveraging entropic independence and a new MLS constant theorem.
• Provide a unified analysis that handles both constant local fields and fields within a constant factor of each other, improving over previous $n^{O(1/\delta)}$ bounds.
• Extend the applicability of spectral independence and MLS-based methods to the interior of the uniqueness regime, including near-critical parameters.

Proposed method

• Develop a new boosting theorem that relates the modified log-Sobolev (MLS) constant of a near-critical Ising model to that of a sub-critical model via entropic independence.
• Utilize the concept of $\alpha$-spectral independence and the Dobrushin influence matrix to bound the MLS constant using the marginal lower bound $\alpha$ and the spectral norm $\|A\|_2$.
• Introduce a flipped distribution $\nu = \text{flip}(\mu, \chi_I)$ to stabilize the marginal bounds and ensure uniform lower bounds on the marginal probabilities across vertices.
• Apply Lemma 8.5 to lower-bound the MLS constant using $\alpha \geq C / (2 \times 10^4)$ and $\|A\|_2 \leq 3/5$, leading to a $\Omega(1/n)$ MLS constant.
• Use the relation $\rho_{\text{GD}}(\pi_\tau) = \frac{m}{n} \rho_{\text{GD}}(\pi_\tau^\Lambda)$ to scale the MLS constant from subgraphs to the full system.
• Establish a tight mixing time bound via $T_{\text{mix}}(\varepsilon) \leq \frac{1}{\rho_{\text{GD}}(\mu)} \left( \log \log \frac{1}{\mu_{\min}} + \log \frac{1}{2\varepsilon^2} \right)$, with $\mu_{\min}$ bounded below in terms of $\lambda_{\min}/\lambda_{\max}$.

Experimental results

Research questions

• RQ1 What is the optimal mixing time of the Glauber dynamics for the Ising model in the uniqueness regime, particularly when the maximum degree $\Delta$ is unbounded?
• RQ2 Can the $n^{O(1/\delta)}$ mixing time bound for $\beta \in \left[ \frac{\Delta-2+\delta}{\Delta-\delta}, \frac{\Delta-\delta}{\Delta-2+\delta} \right]$ be improved to $O(n\log n)$?
• RQ3 How can entropic independence and modified log-Sobolev constants be leveraged to achieve a tight boosting theorem for near-critical Ising models?
• RQ4 Does the mixing time bound depend on the ratio $\lambda_{\max}/\lambda_{\min}$, and if so, how can it be controlled in the analysis?
• RQ5 Can the analysis be extended to yield an $O(n\log^2 n)$ sampling algorithm with $\delta$-dependent constants?

Key findings

• The paper proves an optimal $O(n\log n)$ mixing time for the Glauber dynamics on the Ising model in the uniqueness regime, matching the theoretical lower bound.
• For any $\delta \in (0,1)$, the mixing time is bounded by $T_{\text{mix}}(\varepsilon) \leq C_\delta \cdot \frac{\lambda_{\max}}{\lambda_{\min}} \cdot n \left( \log \frac{n}{\varepsilon} + \log \log \frac{2\lambda_{\max}}{\lambda_{\min}} \right)$, where $C_\delta = \exp(O(1/\delta))$.
• The bound holds for both ferromagnetic and anti-ferromagnetic Ising models on general graphs with unbounded maximum degree $\Delta$.
• The modified log-Sobolev (MLS) constant is shown to be at least $\Omega(1/n)$, which directly implies the $O(n\log n)$ mixing time via standard bounds.
• A refined boosting theorem using entropic independence enables the transfer of MLS bounds from sub-critical to near-critical models, overcoming previous limitations.
• An $O_\delta(n\log^2 n)$ sampling algorithm is shown to be achievable by combining the current results with field dynamics, with constants depending only on $\delta$.