اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Brown measure of unbounded variables with free semicircular imaginary part
91
0
0.0
(
0
)
تأليف
Ching-Wei Ho
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 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.
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous
with a density that is emph{constant along the vertical direction} in the support of $rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $rho_t$ by a natural map is the law of $x_0+s_t$. Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form [frac{1}{r^2}w_t(theta)] in polar coordinates in its support. The support of $mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $mu_t$ by a natural map is the law of $uu_t$. In the special case that $u$ is Haar unitary, the Brown measure $mu_t$ follows the emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by [frac{1}{2pi t}frac{1}{r^2}] in its support.
We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free en
ergy including the negative infinity value of the coupling constant $beta$. Our proof of existence at $beta=-infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $beta=-infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
