A tightness criterion for random fields, with application to the Ising model

TL;DR

This paper introduces a tightness criterion for random fields in certain function spaces and applies it to analyze the critical magnetization field of the 2D Ising model, addressing an open question.

Contribution

It develops a new tightness criterion for random distributions and applies it to the critical Ising model's magnetization field, providing new insights into its regularity.

Findings

Established a tightness criterion in local H"older and Besov spaces.

Applied the criterion to the 2D Ising model at criticality.

Answered an open question by Camia, Garban, and Newman.

Abstract

We present a criterion for a family of random distributions to be tight in local H\"older and Besov spaces of possibly negative regularity. We then apply this criterion to the magnetization field of the two-dimensional Ising model at criticality, answering a question of Camia, Garban and Newman.