Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hariharan Narayanan, Scott Sheffield

Abstract

Suppose $α, β$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $γ$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $α(0) = γ(0) = 0$, and $α(1) = β(0) = 0$ and $β(1) = γ(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $λ= \partial^- α$, $μ= \partial^- β$ on $(0, 1]$ and $ν= \partial^- γ,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $λ_n(i) := n^2(α(\frac{i}{n})-α(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $μ_n(i) := n^2(β(\frac{i}{n})-β(\frac{i-1}{n}))$, and $ν_n(i) := n^2(γ(\frac{i}{n})-γ(\frac{i-1}{n}))$. Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $λ_n$, $μ_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ 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. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - ν_n\|_{\mathcal{I}} < n^2 ε\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $σ$ of continuum limits of the discrete hives defined by Knutson and Tao.

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

Abstract

Suppose are Lipschitz strongly concave functions from to and is a concave function from to , such that , and and For an Hermitian matrix , let denote the vector in whose coordinates are the eigenvalues of listed in non-increasing order. Let , on and at all points of , where is the left derivative, which is monotonically decreasing. Let , for , and similarly, , and . Let be independent random Hermitian matrices from unitarily invariant distributions with spectra , respectively. We define norm to correspond in a certain way to the sup norm of an antiderivative. For suitable and , we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - ν_n\|_{\mathcal{I}} < n^2 ε\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension of continuum limits of the discrete hives defined by Knutson and Tao.

Paper Structure

This paper contains 14 sections, 61 theorems, 308 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation from Representation Theory and Random Matrices
  3. Preliminaries
  4. Large deviations principle
  5. Periodic boundary conditions
  6. Hives
  7. Convex geometry
  8. Volume of the polytope $P_n(s)$
  9. Uniform integrability and submartingale convergence
  10. Lower bound
  11. Upper bound
  12. Large deviations for random hives
  13. Large deviations for the spectrum of the sum of two random matrices
  14. Open problems

Key Result

Lemma 1

Let $h:T \rightarrow \mathbb R$ be a rhombus concave function. Let the restrictions of $h$ to the line segment joining $(0, 0)$ and $(0, 1)$ and to the line segment joining $(0, 1)$ and $(1, 1)$ be $L$-Lipschitz functions. Then $h$ is $CL$-Lipschitz on $T$.

Figures (6)

  • Figure 1.1: Values taken at interior vertices in the hive model
  • Figure 2.1: A visual depiction of the the second order discrete operators ${{\Delta}}_i$. A red dot indicates the vertex $(v_1, v_2)$.
  • Figure 2.2: An augmented hive for $n = 4$. The right triangle below the main diagonal of the $4\times 4$ square corresponds to a random Gelfand-Tsetlin (GT) pattern (printed in red at the center of a diagonal edge) with top row $\nu_n$, given by the difference of the number at a vertex and the one southwest to it. The right triangle above the main diagonal corresponds to a random hive with boundary conditions, $(\lambda_4, \mu_4, \nu_4)$.
  • Figure 3.1: Some translate of $(1 - \frac{C}{n_1^{2}})Q_{\mathbf b_2} (\psi_{\mathbf b_1, \mathbf b_2}(x))$ is contained inside $Q_{\mathbf b_2} (y)$.
  • Figure 5.1: Partition with multiple layers separating the squares and triangles. Dashed lines indicate borders of dyadic pieces.
  • ...and 1 more figures

Theorems & Definitions (161)

  • Definition 1
  • Definition 2
  • Definition 3: Discrete Hessian and the ${{\Delta}}_i$ on $T_n$
  • Definition 4: Rhombus concavity
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 5: Folland, page 212
  • Lemma 3
  • ...and 151 more