Critical threshold for weakly interacting log-correlated focusing Gibbs measures

TL;DR

This paper investigates phase transitions in weakly interacting log-correlated Gibbs measures on a torus, identifying a critical threshold that distinguishes between convergence to Gaussian measures and non-convergence, thus resolving a longstanding open problem.

Contribution

It establishes a critical coupling threshold for log-correlated Gibbs measures, characterizing the transition between weak and strong coupling regimes and answering an open question from 1996.

Findings

In the weak coupling regime, measures converge to a Gaussian measure.

In the strong coupling regime, measures do not converge, even subsequentially.

The critical threshold separates these two regimes.

Abstract

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to $0$ as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).