Polynomial rate of relaxation for the Glauber dynamics of infinite-volume critical Ising model
Haoran Hu
TL;DR
The paper addresses the problem of quantifying the relaxation rate of the infinite-volume critical Ising Glauber dynamics on $\mathbb{Z}^d$ by establishing a polynomial decay for the equal-position spin correlation under two finite-volume assumptions: a polynomially bounded inverse log-Sobolev constant $\gamma_{\Lambda_L}^{-1} \lesssim L^{\eta}$ and a power-law decay of the 1-arm magnetization $\langle\sigma_0\rangle^{+}_{\Lambda_L} \lesssim L^{-\delta}$ with $\delta\in(0,1]$. The approach uses finite-volume approximations on the translation-invariant torus and a time-dependent block-decomposition, leveraging the Bodineau–Helffer inequality and entropy-based methods to bound covariances and propagate bounds to the infinite-volume limit. The main result shows $\langle\sigma_0, P_t \sigma_0\rangle_{\mathbb{Z}^d} \le \frac{C}{t^{\alpha}}$ with $\alpha=\frac{\min(2\delta,1)}{2\eta}$, and establishes a finite-volume analogue on $\mathbb{T}_L$ with constants independent of $L$, thereby demonstrating algebraic relaxation at criticality. This work advances the understanding of critical dynamics by connecting log-Sobolev and entropy techniques with arm-exponent estimates in high dimensions. The findings have implications for predicting mixing times and temporal correlations in critical spin systems. All mathematical notation is kept explicit and rigorously parameterized, reflecting the dependence on $\eta$, $\delta$, and volume via $\alpha$.
Abstract
We consider the relaxation time for the Glauber dynamics of infinite-volume critical ferromagnetic Ising model on $\Z^{d}$ in any dimension $d\geq2$. Under the assumptions regarding the finite-volume log-Sobolev constant and the 1-arm exponent of the critical 1-spin expectation, we show that the equal-position temporal spin correlation function decays polynomially fast in time.