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Register a new userThe Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element
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We compute the Brown measure of $x_{0}+isigma_{t}$, where $sigma_{t}$ 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 $Omega_{t}$ in the plane. In $Omega_{t},$ 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 $x_{0}+isigma_{t}$ by a certain map $Q_{t}:Omega_{t}rightarrowmathbb{R}$ gives the distribution of $x_{0}+sigma_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+isigma_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.
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We compute the Brown measure of the sum of a self-adjoint element and an elliptic element. We prove that the push-forward of this Brown measure of a natural map is the law of the free convolution of the self-adjoint element and the semicircle law; it is also a push-forward measure of the Brown measure of the sum of the self-adjoint element and a circular element by another natural map. We also study various asymptotic behaviors of this family of Brown measures as the variance of the elliptic element approaches infinity.
We consider a family of free multiplicative Brownian motions $b_{s,tau}$ parametrized by a positive real number $s$ and a nonzero complex number $tau$ satisfying $leftvert tau-srightvert leq s,$ with an arbitrary unitary initial condition. We compute the Brown measure $mu_{s,tau}$ of $b_{s,tau}$ and find that it has a simple structure, with a density in logarithmic coordinates that is constant in the $tau$-direction. We also find that all the Brown measures with $s$ fixed and $tau$ varying are related by pushforward under a natural family of maps. Our results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $tau=s.$ We use a version of the PDE method introduced by Driver-Hall-Kemp, but with some significant technical differences.
Let $x_0$ be a possibly-unbounded self-adjoint random variable, $tildesigma_alpha$ and $sigma_beta$ are semicircular variables with variances $alphageq 0$ and $beta>0$ respectively (when $alpha = 0$, $tildesigma_alpha = 0$). Suppose $x_0$, $sigma_alpha$, and $tildesigma_beta$ are all freely independent. We compute the Brown measure of $x_0+tildesigma_alpha+isigma_beta$, extending the previous computations which only work for bounded self-adjoint random variable $x_0$. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+sigma_{alpha+beta}$. We also compute the example where $x_0$ is Cauchy-distributed.
The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $mathsf{GL}(N;mathbb{C}),$ in the sense of $ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_{t}$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $Sigma_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $bar{Sigma}_{t},$ which is strictly positive and real analytic on $Sigma_{t}$. This density has a simple form in polar coordinates: [ W_{t}(r,theta)=frac{1}{r^{2}}w_{t}(theta), ] where $w_{t}$ is an analytic function determined by the geometry of the region $Sigma_{t}$. We show also that the spectral measure of free unitary Brownian motion $u_{t}$ is a shadow of the Brown measure of $b_{t}$, precisely mirroring the relationship between Wigners semicircle law and Ginibres circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.
We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $pm infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.
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