The Gaussian process for particle masses in the near-critical Ising model

TL;DR

This paper connects the near-critical Ising model's magnetization to a Gaussian process derived from quantum field theory, demonstrating convergence of the renormalized magnetization to this process and discussing potential extensions to higher dimensions.

Contribution

It constructs a Gaussian process from the near-critical Ising model and proves the convergence of the renormalized magnetization to this process in two dimensions.

Findings

The renormalized magnetization converges to the Gaussian process $X(t)$ as lattice spacing goes to zero.

The Gaussian process $X(t)$ is related to the continuum scaling limit of the Ising magnetization.

Potential extension of the approach to higher dimensions is discussed.

Abstract

We review the construction of a stationary Gaussian process $X(t)$ starting from the near-critical continuum scaling limit $\Phi^h$ of the Ising magnetization and its relation to the mass spectrum of the relativistic quantum field theory associated to $\Phi^h$. Then for the near-critical Ising model on $a \mathbb{Z}^2$ with external field $a^{15/8} h$, we study the renormalized magnetization along a vertical line (with horizontal coordinate approximately $t$) and prove that the limit as $a\downarrow 0$ is the same Gaussian process $X(t)$. We also explore the possible extension of this approach to dimensions $d > 2$.