Large deviations for random hives and the spectrum of the sum of two random matrices

Abstract

Suppose α, β are Lipschitz, strongly concave functions from [0,1] to R and γ is a concave function from [0,1] to R such that α(0)=γ(0)=0, α(1)=β(0)=0 and β(1)=γ(1)=0. For an n×n Hermitian matrix W, let spec(W) denote the vector in Rn whose coordinates are the eigenvalues of W listed in nonincreasing order. Let λ=∂−α, μ=∂−β on (0,1] and ν=∂−γ, at all points of (0,1], where ∂− is the left derivative. Let λn(i):=n2(α(in)−α(i−1n)), for i∈[n], and similarly, μn(i):=n2(β(in)−β(i−1n)) and νn(i):=n2(γ(in)−γ(i−1n)). Let Xn, Yn be independent random Hermitian matrices from unitarily invariant distributions with spectra λn, μn, respectively. We define norm ‖·‖I to correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists: limn→∞ logP[‖spec(Xn+Yn)−νn‖I<n2ϵ] n2. We interpret this limit in terms of the surface tension σ of continuum limits of the discrete hives defined by Knutson and Tao. We provide matching large deviation upper and lower bounds for the spectrum of the sum of two random matrices Xn and Yn, in terms of the surface tension σ mentioned above. We also prove large deviation principles for random hives with α and β that are C2, where the rate function can be interpreted in terms of the maximizer of a functional that is the sum of a term related to the free energy of hives associated with α, β and γ and a quantity related to logarithms of Vandermonde determinants associated with γ.

Locations

• The Annals of Probability - View
• arXiv (Cornell University) - View - PDF