Operator limit of Wigner matrices I

2025/03/31 by Banerjee, Debapratim
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2503.23940

Abstract

We consider the Wigner matrix W_n of dimension n × n as n → ∞. The objective of this paper is two folds: first we construct an operator W on a suitable Hilbert space H and then define a suitable notion of convergence such that the matrices W_n converge in that notion of convergence to W. We further investigate some properties of W and H. We show that H is a nontrivial extension of L²[0,1] with respect to the Lebesgue measure and the spectral measure of W at any function f ∈ L²[0,1] is almost surely the semicircular law.

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