Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models

TL;DR

This paper analyzes hierarchical integer-valued Gaussian and Coulomb gas models, deriving asymptotic formulas for correlations and fractional charges across different temperature regimes, revealing phase transitions and critical behavior.

Contribution

It provides the first sharp asymptotic analysis of covariance and fractional charge behavior in hierarchical models near criticality using renormalization group techniques.

Findings

Logarithmic correlations are present throughout all regimes.

Distinct $eta$-dependence of covariance and fractional-charge exponents.

Explicit logarithmic corrections are identified at the critical point.

Abstract

Given a square box $\Lambda_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{\phi_x\colon x\in\Lambda_n\}$ with law proportional to ${\rm e}^{\frac12\beta(\phi,\Delta_n\phi)}\prod_{x\in\Lambda_n}\nu({\rm d}\phi_x)$, where $\beta>0$ is the inverse temperature, $\Delta_n$ is a hierarchical Laplacian on $\Lambda_n$, and $\nu$ is a non-degenerate $1$-periodic measure on $\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as $n\to\infty$, for the covariance $\langle\phi_x\phi_y\rangle$ and the fractional charge $\langle {\rm e}^{2\pi {\rm i}\alpha(\phi_x-\phi_y)}\rangle$ in the subcritical $\beta<\beta_{\rm c}:=\pi^2/\log L$, critical $\beta=\beta_{\rm c}$ and slightly supercritical $\beta>\beta_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $\beta$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.