A limit law for the maximum of subcritical DG-model on a hierarchical lattice

Abstract

We study the extremal properties of the "integer-valued Gaussian" a.k.a.\ DG-model on the hierarchical lattice $\Lambda_n:=\{1,\dots,b\}^n$ (with $b\ge2$) of depth $n$. This is a random field $\varphi\in\mathbb Z^{\Lambda_n}$ with law proportional to $e^{\frac12\beta(\varphi,\Delta_n\varphi)}\prod_{x\in\Lambda_n}\#(d\varphi_x)$, where $\Delta_n$ is the hierarchical Laplacian, $\beta$ is the inverse temperature and $\#$ is the counting measure on $\mathbb Z$. Denoting $\beta_c:=2\pi^2/\log b$ and $m_n:=\beta^{-1/2}[(2\log b)^{1/2}n-\frac32(2\log b)^{-1/2}\log n]$, for $0<\beta<\beta_c$ we prove that, along increasing sequences of $n$ such that the fractional part of $m_{n}$ converges to an $s\in[0,1)$, the centered maximum $\max_{x\in\Lambda_n}\varphi_x-\lfloor m_n\rfloor$ tends (as $n\to\infty$) in law to a discrete variant of a randomly shifted Gumbel law with the shift depending non-trivially on $s$. The convergence extends to the extremal process whose law tends to a decorated Poisson point process with a random intensity measure. The proofs rely on renormalization-group analysis which enables a tight coupling of the DG-model to a Gaussian Free Field. The interval $(0,\beta_c]$ marks the full range of values of $\beta$ for which the renormalization-group iterations tend to a "trivial" fixed point.