LDP for the largest eigenvalue of Kronecker random matrices

TL;DR

The paper establishes a full large deviation principle at speed N for the largest eigenvalue of Gaussian Kronecker random matrices ${\bf X}_\beta^{(N)}=\sum_{j=1}^k A_j\otimes W_j^{(N)}+A_0\otimes \mathrm{Id}_N$, in the regime where the Gaussian blocks grow large. The authors develop a tilting method via spherical integrals, analyze annealed spherical integrals, and track a top-eigenvector profile through a Matrix Dyson Equation framework, yielding a variational rate function that couples the base variance structure with a renormalized Wishart profile. They prove both weak upper and lower bounds, and show the rate function is good, increasing, and continuous on (r∞,∞) with possible discontinuity at r∞; in the complex case (GUE) the bounds adapt with Haagerup–Thorbjørnsen machinery and satisfy I2(x) ≤ 2 I1(x). The work provides the first large deviation principle for matrices with correlated entries in a Kronecker-structured model, and highlights connections to spectral entropy concepts in free probability and potential extensions to polynomials in random matrices.

Abstract

We prove a large deviations principle for the largest eigenvalue of Gaussian Kronecker matrices, namely matrices defined as the sum of tensors of independent Gaussian matrices in the regime where the dimension of the Gaussian matrices goes to infinity.

Theorems & Definitions (49)

• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Weak LDP
• Lemma 2.2: GuionnetH22
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 4.1
• proof : Proof of Lemma \ref{['lem-optimizationreg']}