The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element

Abstract

We compute the Brown measure of x 0 + iσt, where σt is a free semicircular Brownian motion and x 0 is a freely independent self-adjoint element. The Brown measure is supported in the closure of a certain bounded region Ωt in the plane. In Ωt, the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature.We also show that pushing forward the Brown measure of x 0 + iσt by a certain map Qt : Ωt → R gives the distribution of x 0 + σt. We also establish a similar result relating the Brown measure of x 0 + iσt to the Brown measure of x 0 + ct, where ct is the free circular Brownian motion.

Brown measure support and the free multiplicative Brownian motion

“…Theorem 4.10 of [23] then gives the alternative characterization of Σ s in (3.6) and shows that T > s outside the closure of Σ s . That points outside the closure of Σ s are outside the closed support of µ 0 follows by the same argument as in the proof of Lemma 6.3 of [20].…”

“…We now briefly summarize the way the computation of the Brown measure "in the domain" (that is, where it is not zero) works in the present paper, since this is the main way our paper differs from earlier ones. In the present paper, as in earlier works such as [13], [23], [10], and [20], one initially attempts to achieve the desired condition ε(t) = 0 by taking ε 0 = 0, leading to a determination of the region where the Brown measure is zero. But the way the idea of letting ε 0 tend to zero fails to work is different here than in the previous works.…”

“…We will therefore use an indirect approach, similar to the one developed in [23,Lemma 4.25] and used also in [20,Theorem 7.9]. In Section 8, we will consider a map Ψ s,τ -denoted there as Φ s • Φ −1 s,τ -from Σ s,τ to the unit circle.…”

“…The method in [13] was then extended by the second author and Zhong [23] to analyze the standard free multiplicative Brownian motion with arbitrary unitary initial condition (τ = s and u 0 is a arbitrary). The method has also been used in [10] and [20] and discussed in the physics literature in [16].…”

The Brown measure of unbounded variables with free semicircular imaginary part

“…Based on this PDE method, Zhong and the author [16] computed the Brown measures of the circular Brownian motion with arbitrary bounded self-adjoint initial condition x 0 and the free multiplicative Brownian motion with arbitrary unitary initial condition. Later, Hall and the author [13] computed the Brown measure of the sum of an arbitrary bounded self-adjoint random variable x 0 and an imaginary multiple of semicircular variable σ t , freely independent from x 0 . Using the results in [13], the author [15] computed the Brown measure of the sum of a bounded self-adjoint random variable x 0 and an elliptic variable c α,β that is freely independent from x 0 .…”