2021/10/31 by Hariharan Narayanan, Scott Sheffield⋆, Narayanan, Hariharan +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.00421
openalex publication_date 2021/10/31 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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, and 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 non-increasing order. Let λ= ∂- α, μ= ∂- β on (0, 1] and ν= ∂- γ, at all points of (0, 1], where ∂- is the left derivative, which is monotonically decreasing. Let λn(i) := n2(α((i)/(n))-α((i-1)/(n))), for i ∈ [n], and similarly, μn(i) := n2(β((i)/(n))-β((i-1)/(n))), and νn(i) := n2(γ((i)/(n))-γ((i-1)/(n))). 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. For suitable λ and μ, we prove that the following limit exists. limn → ∞\fracln ℙ[‖spec(Xn + Yn) - νn‖I lt; n2 ε]n2. We interpret this limit in terms of the surface tension σ of continuum limits of the discrete hives defined by Knutson and Tao.