On a class of random perturbations of the hierarchical Laplacian

A. D. Bendikov ; A. A. Grigor'yan ; S. A. Molchanov ; G. P. Samorodnitsky

Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 859-893

Résumé

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set $B$ of all balls of positive measure of $X$, we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts on $L^{2}(X,m)$. It is essentially self-adjoint and has a pure point spectrum. By choosing a family $\{\varepsilon (B)\}$ of independent identically distributed random variables, we define the perturbed function $C(B,\omega)$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega)}$. We study the arithmetic means $\bar{\lambda }(\omega)$ of the eigenvalues of $L^{\omega }$. Under some mild assumptions the normalized arithmetic means $( \bar{\lambda }-\mathbb{E}\bar{\lambda })/\sigma [\bar{\lambda }]$ converge to $N(0,1)$ in distribution. We also give examples when the normal convergence fails. We prove the existence of an integrated density of states. Introducing an empirical point process $N^{\omega }$ for the eigenvalues of $L^{\omega }$ and assuming that the density of states exists and is continuous, we prove that the finite-dimensional distributions of $N^{\omega }$ converge to those of the Poisson point process. As an example we consider random perturbations of the Vladimirov operator acting on $L^{2}(X,m)$, where $X=\mathbb{Q}_{p}$ is the ring of $p$-adic numbers and $m$ is the Haar measure.

Détail

Keywords: ultrametric measure space, field of $p$-adic numbers, hierarchical Laplacian, fractional derivative, Vladimirov Laplacian, point spectrum, integrated density of states, Erdős problem, point process
Mots-clés : Bernoulli convolutions, Poisson convergence.