The Brown measure of a family of free multiplicative Brownian motions

Abstract

We consider a family of free multiplicative Brownian motions $b_{s,\tau }$ parametrized by a real variance parameter $s$ and a complex covariance parameter $\tau .$ We compute the Brown measure ${\mu }_{s,\tau }$ of $u{b}_{s,\tau },$ where $u$ is a unitary element freely independent of $b_{s,\tau }.$ We find that ${\mu }_{s,\tau }$ has a simple structure, with a density in logarithmic coordinates that is constant in the $\tau$-direction. These results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $\tau =s.$ We also establish a remarkable "model deformation phenomenon," stating that all the Brown measures with $s$ fixed and $\tau$ varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton-Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver-Hall-Kemp, our methods are substantially different at both the technical and conceptual level.

Author final version. To appear in Probability Theory and Related Fields. 72 pages with 14 figures

References in corpus (3)

• The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element
• Eikonal formulation of large dynamical random matrix models
• PDE methods in random matrix theory

Cited by in corpus (2)

• The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element
• Zeros of random polynomials undergoing the heat flow