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

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.