The local Poincare inequality of stochastic dynamic and application to the Ising model

TL;DR

This paper develops a local Poincaré inequality for stochastic dynamics inspired by stochastic quantization, providing new tools to analyze correlation functions in the Ising model through probabilistic and geometric methods.

Contribution

It introduces a novel approach to establish a local Poincaré inequality for infinite-dimensional stochastic dynamics related to the Ising model, connecting stochastic analysis with renormalization techniques.

Findings

Established local Poincaré inequality for stochastic dynamics

Derived estimates for correlation functions in the Ising model

Linked ergodicity conditions to correlation decay

Abstract

Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we construct the transition probability matrix which plays a central role in the renormalization group through a stochastic differential equation. By establishing the discrete time stochastic dynamics, the renormalization procedure can be characterized from the perspective of probability. Hence, we will focus on the investigation of the infinite dimensional stochastic dynamic. From the stochastic point of view, the discrete time stochastic dynamic can induce a Markov chain. Via calculating the square field operator and the Bakry-\'Emery curvature for a class of two-points functions, the local Poincar\'e inequality is established, from which the estimate of correlation functions can also be obtained. Finally, under the condition of ergodicity, by choosing the couple relationship between the system parameter $K$ and the system time $T$ properly when $T\rightarrow +\infty$, the two-points correlation functions for limit system are also estimated.