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 $b\ge 2$) of depth $n$. This is a random field $\phi \in {\mathbb{Z}}^{{\Lambda }_n}$ with law proportional to $e^{\frac{1}{2}\beta (\phi ,{\Delta }_n\phi )}{\prod }_{x\in {\Lambda }_n}\#(d{\phi }_x)$, 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 $m_n:={\beta }^{-1/2}[(2\log b{)}^{1/2}n-\frac{3}{2}(2\log b{)}^{-1/2}\log n]$, 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 ${\max }_{x\in {\Lambda }_n}{\phi }_x-\lfloor m_n\rfloor$ 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.

53 pages