Matrix random products with singular harmonic measure

\random" Random Matrix Products

Singular Sets for Harmonic Measure on Locally Flat Domains with Locally Finite Surface Measure

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each d > 1, there exists a Reifenberg flat domain Ω ⊂ R with H(∂Ω) < ∞ and a subset E ⊂ ∂Ω with positive harmonic measur...

Singular value correlation functions for products of Wishart random matrices

We consider the product ofM quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalises the classical Wishart-Laguerre Gaussian Unitary Ensemble with M = 1. In this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix sizeN and t...

The harmonic measure of balls in random trees

We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius n centered at the root, we prove that, although the size of the boundary is of order n, most of the harmonic measure is supported on a boundary set of size approximately equal to n , where β ≈ 0.78 is a universal constant. To derive such results, we interpret harmonic measure as the exit d...

Harmonic Maps with Fixed Singular Sets

§ 0. Introduction Here, for a smooth domain Ω in R and a compact smooth Riemannian manifold N we study a space H consisting of all harmonic maps u : Ω → N that have a singular set being a fixed compact subset Z of Ω having finite m− 3 dimensional Minkowski content. This holds if, for example, Z is m− 3 rectifiable [F, 3.2.14]. We define a suitable topology on H using Hölder norms on derivatives...