Poisson statistics of eigenvalues in the hierarchical Dyson model

Abstract

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C$ defined on the set $\mathcal{B}$ of all balls $B\subset X$ we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts in $L^{2}(X,m)$, is essentially self-adjoint, and has a purely point spectrum. Choosing a family $\{\varepsilon(B)\}_{B\in \mathcal{B}}$ of i.i.d. random variables, we define the perturbed function $\mathcal{C}(B)=C(B)(1+\varepsilon(B))$ and the perturbed hierarchical Laplacian $\mathcal{L}=L_{\mathcal{C}}$. All outcomes of the perturbed operator $\mathcal{L}$ are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process $M$ defined in terms of $\mathcal{L}$-eigenvalues. Under some natural assumptions $M$ can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen-Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator $\mathfrak{D}^{\alpha }$, the $p$-adic fractional derivative of order $\alpha >0$. This operator, related to the concept of $p$-adic Quantum Mechanics, is a hierarchical Laplacian which acts in $L^{2}(X,m)$ where $X=\mathbb{Q}_{p}$ is the field of $p$-adic numbers and $m$ is Haar measure. It is translation invariant and the set $\mathsf{Spec}(\mathfrak{D}^{\alpha })$ consists of eigenvalues $p^{\alpha k}$, $k\in \mathbb{Z}$, each of which has infinite multiplicity.