Large deviations for random hives and the spectrum of the sum of two random matrices
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Author(s)
Narayanan, Hariharan
Sheffield, Scott
Date Issued
May 2024
Journal
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Citation
Hariharan Narayanan. Scott Sheffield. "Large deviations for random hives and the spectrum of the sum of two random matrices." Ann. Probab. 52 (3) 1093 - 1152, May 2024.
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Final published version
Abstract
Suppose α, β are Lipschitz, strongly concave functions from [0ⓜ,1]to Rand γ is a concave function from [0ⓜ,1]to Rsuch that α(0)=γ(0)=0α(1)=β(0)=0and β(1)=γ(1)=0n×nHermitian matrix W, let spec(W)denote the vector in R^nwhose coordinates are the eigenvalues of W listed in nonincreasing order.
Let λ=∂^- αμ=∂^- βon (0ⓜ,1]and ν=∂^- γ(0ⓜ,1] ∂^-is the left derivative. Let λ_n (i):=n^2 (α(i/n)-α((i-1)/n))i∈[n] μ_n (i):=n^2 (β(i/n)-β((i-1)/n))and ν_n (i):=n^2 (γ(i/n)-γ((i-1)/n)) X_n Y_nbe independent random Hermitian matrices from unitarily invariant distributions with spectra λ_n μ_n ‖·‖_Ito correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists:
lim┬(n→∞) (logP[specⓜ‖(X_n+Y_n )-ν_n ├ ‖┤_I
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DOI of Published Version
https://doi.org/10.1214/24-aop1687
