The log-Sobolev inequality and correlation functions for the renormalization of 1D Ising model
Kaiyuan Cui, Fuzhou Gong
TL;DR
The paper tackles renormalization for the 1D Ising model through a stochastic quantization inspired SPDE obtained by mollifying and continuumizing a discrete equation. It develops a rigorous regularity theory, derives a Clark-Ocone-Haussmann representation and a log-Sobolev inequality up to a terminal time $T$, and uses these to define a renormalization relation via a partition-function argument. The main contributions are the construction of a renormalization that keeps the log-Sobolev constant controlled as $T$ grows, the proof of partition function invariance, and the demonstration that lattice two-point correlations converge to those of the 1D Ising model at the stable RG fixed point as $T\to\infty$. This provides a principled, probabilistic route to RG analysis for a fundamental statistical mechanics model and suggests a pathway to extend the approach to more complex field theories.
Abstract
The renormalization group (RG) method is an important tool for studying critical phenomena. In this paper, we employ stochastic analysis techniques to investigate the stochastic partial differential equation (SPDE) derived by regularizing and continuousizing the discrete stochastic equation, which is a variant of stochastic quantization equation of the one dimensional (1D) Ising model. Firstly, we give the regularity estimates for the solution to SPDE. Secondly, we prove the Clark-Ocone-Haussmann formula and derive the log-Sobolev inequality up to the terminal time $T$, as well as obtain a priori form of the renormalization relation. Finally, we verify the correctness of the renormalization procedure based on the partition function, and prove that the two point correlation functions of SPDE on lattices converge to the two point correlation functions of the 1D Ising model at the stable fixed point of the RG transformation as $T\rightarrow +\infty$.