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Dr. Alexander Bendikov, Department of Mathematics, Cornell University
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Abstract: Let (X,d) be a locally compact separable metric space. We assume that (X,d) is proper and totally disconnected. Given a measure m on X and a choice-function C(B) defined on the set of all non-singleton balls B of X we consider the hierarchical Laplacian L=L_{C}. The operator L acts in L(X,m), is essentially self-adjoint and has a purely point spectrum. Choosing a sequence of i.i.d. random variables we define a perturbated choice-function and a perturbated hierarchical Laplacian. We study asymptotic behaviour of the arithmetic means of the eigenvalues of the perturbated hierarchical Laplacian. We prove that under some mild assumptions the arithmetic means are asymptotically normal whereas without these conditions the limited distribution is not normal. We prove existence of the integrated density of states (i.d.s.) associated with the operator perturbated hierarchical Laplacian – the quantitative characteristics of the phase transition in the Dyson’s model. Assuming that the i.d.s is continuous we prove that the number of eigenvalues which fall in a small interval is approximately Poissonian. As an example we consider random perturbations defined by the i.i.d. Bernoulli random variables of the Vladimirov-Taibleson operator acting on the ring of p-adic numbers.
This is joint work with A. Grigor’yan and S.A. Molchanov.
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Thursday, April 3 at 3:30pm in Room Fretwell 406
Categories: Spring 2022