Local limit theorems for multiplicative free convolutions

Abstract

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici-Pata bijection.

Superconvergence to freely infinitely divisible distributions

“…As the parameter δ can be arbitrarily small, we have proved the uniform convergence of s n → s on I ′ . This finishes the proof of Part (1).…”

“…Even when the variables X i are not identically distributed, but are uniformly bounded, the support of S n was shown by Kargin [12] to converge to the interval [−2, 2] as n → ∞. See also [1] for free multiplicative superconvergence results. The purpose of this paper is to demonstrate that the phenomenon of superconvergence is not limited to convergence to the semicircle law.…”

“…Theorem 1. 1. Given any open set U ⊃ D ν , the distribution ν n of S n is absolutely continuous on R\U for sufficiently large n, and the density of ν n converges to the density of ν uniformly and in L p -norms for p > 1 on R\U .…”

Matricial model for the free multiplicative convolution

“…We would point out the recent work [AWZ13] which proves that the convergence of Theorem 3.8 implies local convergences of the probability densities.…”

Limits of Radial Multiple SLE and a Burgers–Loewner Differential Equation

“…The case S µ (z) = 2z describes the infinite-slit limit of radial multiple SLE with equal weights. The measures from (A) in this special case describe the distributions of a (time changed) free unitary Brownian motion, see [AWZ14]. (In particular, see [AWZ14, p. 3490], which gives the Σ-transforms of these measures.…”