Collision of eigenvalues for matrix-valued processes

TL;DR

This paper investigates the conditions under which eigenvalues of Hermitian matrix-valued Gaussian processes collide, revealing sharp thresholds related to the Hurst parameter for fractional Brownian motions.

Contribution

It establishes precise criteria for eigenvalue collisions in matrix-valued Gaussian processes, linking collision probabilities to Gaussian process hitting probabilities and geometric measures.

Findings

Eigenvalues of real symmetric fractional Brownian motion collide when H<1/2.

Eigenvalues of complex Hermitian fractional Brownian motion collide when H<1/3.

Eigenvalues do not collide when H exceeds these thresholds.

Abstract

We examine the probability that at least two eigenvalues of an Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter $H$, collide when $H<1/2$ and don't collide when $H>\frac{1}{2}$, while those of a complex Hermitian fractional Brownian motion collide when $H<\frac{1}{3}$ and don't collide when $H>\frac{1}{3}$. Our approach is based on the relation between hitting probabilities for Gaussian processes with the capacity and Hausdorff dimension of measurable sets.