Rigorous lower bound of the dynamical critical exponent of the Ising model

TL;DR

This paper addresses the problem of determining the dynamical critical exponent $z$ for the kinetic Ising model under Glauber dynamics by linking relaxation to the spectral gap of the finite-volume Markov generator. It introduces a mapping to a frustration-free Rokhsar-Kivelson Hamiltonian and applies correlation inequalities of Simon-Lieb and Gosset-Huang to bound correlation functions. The main result is a rigorous finite-volume bound $0<\epsilon_L \le C\left(\frac{\log L}{L}\right)^2$ for $\beta\ge\beta_c$, which implies $z\ge 2$ through standard finite-size scaling since $\epsilon_L \sim L^{-z}$ at criticality. This strengthens prior bounds and provides a dimension-independent, rigorous constraint on dynamical critical slowing down with potential applicability to broader Markov processes and other models in equilibrium statistical mechanics.

Abstract

We study the kinetic Ising model under Glauber dynamics and establish an upper bound on the spectral gap for finite systems. This bound implies the critical exponent inequality $z \geq 2$, thereby rigorously improving the previously known estimate $z \geq 2 - η$. Our proof relies on the mapping from stochastic processes to frustration-free quantum systems and leverages the Simon--Lieb and Gosset--Huang inequalities.