Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models
Marek Biskup, Haiyu Huang
TL;DR
This work analyzes Z-modulated hierarchical interfaces on $\mathbb Z^2$ and establishes a sharp, universal picture of phase transitions driven by the hierarchical Laplacian. By recasting the Gibbs measure as a tree-indexed Markov chain and tracking the full Fourier-coefficient flow $\{a_k(q)\}$ through a robust RG framework, the authors derive precise subcritical, critical, and near-critical asymptotics for the covariance and fractional-charge correlations, including logarithmic corrections at criticality. A key novelty is the use of the exponential of the renormalized potentials to achieve uniform control and universality across a broad class of initial measures $\nu$ (including the DG and sine-Gordon cases) under mild assumptions. In the supercritical regime, the paper identifies a nontrivial RG fixed point, proves existence/uniqueness of the fixed point, and demonstrates contraction toward it in a contractive region, yielding a detailed near-critical description of the RG flow and its implications for observables. Overall, the results provide a comprehensive RG-driven account of BKT-type behavior in hierarchical 2D Coulomb-gas/CFT-type models with logarithmic correlations and explicit critical-exponent structures.
Abstract
Given a square box $Λ_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{φ_x\colon x\inΛ_n\}$ with law proportional to ${\rm e}^{\frac12β(φ,Δ_nφ)}\prod_{x\inΛ_n}ν({\rm d}φ_x)$, where $β>0$ is the inverse temperature, $Δ_n$ is a hierarchical Laplacian on $Λ_n$, and $ν$ 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 $\langleφ_xφ_y\rangle$ and the fractional charge $\langle {\rm e}^{2π{\rm i}α(φ_x-φ_y)}\rangle$ in the subcritical $β<β_{\rm c}:=π^2/\log L$, critical $β=β_{\rm c}$ and slightly supercritical $β>β_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $β$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.