Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

  • Ching-Wei Ho

    Academia Sinica, Taipei, Taiwan; University of Notre Dame, USA
  • Ping Zhong

    University of Wyoming, Laramie, USA
Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions cover
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Abstract

Let be a Ginibre ensemble and let be a Hermitian random matrix independent of such that converges in distribution to a self-adjoint random variable in a -probability space . For each , the random matrix converges in -distribution to , where is a circular variable of variance , freely independent of . We use the Hamilton–Jacobi method to compute the Brown measure of . The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of is related to the subordination function of the free additive convolution of , where is a semicircular variable of variance , freely independent of . Furthermore, the push-forward of by a natural map is the law of .

Let be the Brownian motion on the general linear group and let be a unitary random matrix independent of such that converges in distribution to a unitary random variable in . The random matrix converges in -distribution to where is the free multiplicative Brownian motion, freely independent of . We compute the Brown measure of , extending the recent work by Driver–Hall–Kemp, which corresponds to the case . The measure has a density of the special form

in polar coordinates in its support. The support of is related to the subordination function of the free multiplicative convolution of where is the free unitary Brownian motion, freely independent of . The push-forward of by a natural map is the law of .

In the special case that is Haar unitary, the Brown measure follows the annulus law . The support of the Brown measure of is an annulus with inner radius and outer radius . In its support, the density in polar coordinates is given by

Cite this article

Ching-Wei Ho, Ping Zhong, Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions. J. Eur. Math. Soc. 25 (2023), no. 6, pp. 2163–2227

DOI 10.4171/JEMS/1233