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- W2228044280 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 $Bsubset X$ 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 ${varepsilon(B)}_{Bin mathcal{B}}$ 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 $p^{alpha k}$, $kin mathbb{Z}$, each of which has infinite multiplicity." @default.
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- W2228044280 date "2018-01-01" @default.
- W2228044280 modified "2023-10-18" @default.
- W2228044280 title "Poisson statistics of eigenvalues in the hierarchical Dyson model" @default.
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- W2228044280 doi "https://doi.org/10.4213/tvp5136" @default.
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