Poisson Statistics of Eigenvalues in the Hierarchical Dyson Model


Let (X, d) be a locally compact separable ultrametric space. Given a measure m on X and a function C defined on the set B of all balls B ⊂ X we consider the hierarchical Laplacian L = LC . The operator L acts in L (X,m), is essentially self-adjoint, and has a purely point spectrum. Choosing a family {ε(B)}B∈B of i.i.d. random variables, we define the perturbed function C(B) = C(B)(1 + ε(B)) and the perturbed hierarchical Laplacian L = LC . All outcomes of the perturbed operator L are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process M defined in terms of 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 D, the p-adic fractional derivative of order α > 0. This operator, related to the concept of p-adic Quantum Mechanics, is a hierarchical Laplacian which acts in L(X,m) where X = Qp is the field of p-adic numbers and m is Haar measure. It is translation invariant and the set Spec(D) consists of eigenvalues p, k ∈ Z, each of which has infinite multiplicity.

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@inproceedings{BENDIKOV2015PoissonSO, title={Poisson Statistics of Eigenvalues in the Hierarchical Dyson Model}, author={ALEXANDER BENDIKOV and John Pike}, year={2015} }