arXiv subjects

Ching-Wei Ho

Publications and source records attributed to Ching-Wei Ho.

At least 19 recordsLinked to original sources

arXiv2025

Spectral results for free random variables

Let $(\mathcal{A},\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\mathrm{tr}:\mathcal{A}\rightarrow\mathbb{C}.$ For each $a\in\mathcal{A},$ define \[ S(\lambda,\varepsilon)=\mathrm{tr}[\log((a-\lambda)^{\ast}(a-\lambda )+\varepsilon)],\quad\lambda\in\mathbb{C},~\varepsilon>0, \] so that the limit as $\varepsilon\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $\lambda\in\mathbb{C},$ the function \[ \varepsilon\mapsto\frac{\partial S}{\partial\varepsilon}(\lambda ,\varepsilon)=\mathrm{tr}[((a-\lambda)^{\ast}(a-\lambda)+\varepsilon )^{-1}] \] admits a real analytic extension to a neighborhood of $0$ in $\mathbb{R}.$ Then we will show that $\lambda$ is outside the spectrum of $a.$ We will apply this result to several examples involving circular and elliptic elements, as well as free multiplicative Brownian motions. In most cases, we will show that the spectrum of the relevant element $a$ coincides with the support of its Brown measure.

math.OA↗

arXiv2025

Outlier eigenvalues for full rank deformed single ring random matrices

Let $A_n$ be an $n \times n$ deterministic matrix and $\Sigma_n$ be a deterministic non-negative matrix such that $A_n$ and $\Sigma_n$ converge in $*$-moments to operators $a$ and $\Sigma$ respectively in some $W^*$-probability space. We consider the full rank deformed model $A_n + U_n \Sigma_n V_n,$ where $U_n$ and $V_n$ are independent Haar-distributed random unitary matrices. In this paper, we investigate the eigenvalues of $A_n + U_n\Sigma_n V_n$ in two domains that are outside the support of the Brown measure of $a +u \Sigma$. We give a sufficient condition to guarantee that outliers are stable in one domain, and we also prove that there are no outliers in the other domain. When $A_n$ has a bounded rank, the first domain is exactly the one outside the outer boundary of the single ring, and the second domain is the inner disk of the single ring. Our results generalize the results of Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 2016).

math.PR↗

arXiv2024

On the support of free convolutions

We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on \mathbb{R}_+. More precisely, the convolution of two measures with connected supports also has connected support. The result holds without any absolute continuity or bounded support hypotheses on the measures being convolved. We also show that the results of Moreillon and Schnelli concerning the number of components of the support of a free additive convolution hold for arbitrary measures with bounded supports. Finally, we provide an approach to the corresponding results in the case of free multiplicative convolutions of probability measures on the unit circle.

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arXiv2023

Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a} (d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the limiting root distribution $\mu_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $\mu_{t}$ is the push-forward of the limiting root distribution of $P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing along the characteristic curves of a PDE satisfied by the log potential of $\mu_{t}.$ In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially \textit{with constant speed} until they hit the origin, at which point, they cease to exist. For general $a$ and $b,$ the transport map $T_{t}$ has a free probability interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal \textquotedblleft transport operator.\textquotedblright As an application, we obtain a push-forward characterization of the free self-convolution semigroup $\oplus$ of radial measures on $\mathbb{C}$. We also consider the case $b<0,$ which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.

math.PR↗

arXiv2023

Zeros of random polynomials undergoing the heat flow

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law, as was recently conjectured for characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover, for a general family of random polynomials with independent coefficients and isotropic limiting distribution of zeros, we determine the zero distribution of the heat-evolved polynomials in terms of its logarithmic potential. Furthermore, we explicitly identify two critical time thresholds, at which singularities develop and at which the limiting distribution collapses to the semicircle law. We completely characterize the limiting root distribution of the heat-evolved polynomials before singularities develop as the push-forward of the initial distribution under a transport map. Finally, we discuss the results from the perspectives of partial differential equations (in particular Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free probability. The theory is accompanied by explicit examples, simulations, and conjectures.

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arXiv2023

The heat flow, GAF, and SL(2;R)

We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow to a GAF and then rescaling and multiplying by an exponential of a quadratic function gives another GAF. It follows that the zeros of the GAF are invariant in distribution under the heat flow, up to a simple rescaling. We then show that the zeros of the GAF evolve under the heat flow approximately along straight lines, with an error whose distribution is independent of the starting point. Finally, we connect the heat flow on the GAF to the metaplectic representation of the double cover of the group $SL(2;\mathbb{R}).$

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arXiv2022

Deformed single ring theorems

Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $\Sigma=\Sigma_N$ such that $A\to a$ and $\Sigma\to \sigma$ in $\ast$-distribution for some operators $a$ and $\sigma$ in a finite von Neumann algebra $\mathcal{A}$. Let $U =U_N$ and $V=V_N$ be independent Haar-distributed unitary matrices. We use free probability techniques to prove that, under mild assumptions, the empirical eigenvalue distribution of $U\Sigma V^*+A$ converges to the Brown measure of $T+a$, where $T\in\mathcal{A}$ is an $R$-diagonal operator freely independent from $a$ and $\vert T\vert$ has the same distribution as $\sigma$. The assumptions can be removed if $A$ is Hermitian or unitary. By putting $A= 0$, our result removes a regularity assumption in the single ring theorem by Guionnet, Krishnapur and Zeitouni. We also prove a local convergence on optimal scale, extending the local single ring theorem of Bao, Erd\H{o}s and Schnelli.

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arXiv2022

The Brown measure of a family of free multiplicative Brownian motions

We consider a family of free multiplicative Brownian motions $b_{s,τ}$ parametrized by a real variance parameter $s$ and a complex covariance parameter $τ.$ We compute the Brown measure $μ_{s,τ}$ of $ub_{s,τ},$ where $u$ is a unitary element freely independent of $b_{s,τ}.$ We find that $μ_{s,τ}$ has a simple structure, with a density in logarithmic coordinates that is constant in the $τ$-direction. These results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $τ=s.$ We also establish a remarkable "model deformation phenomenon," stating that all the Brown measures with $s$ fixed and $τ$ varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton-Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver-Hall-Kemp, our methods are substantially different at both the technical and conceptual level.

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arXiv2022

The Brown measure of the sum of a self-adjoint element and an elliptic element

We completely determine the Brown measure of the sum of a self-adjoint element and an elliptic element, which is the limiting eigenvalue distribution of the random matrix \[Y_N+\sqrt{s-\frac{t}{2}}X_N+i\sqrt{\frac{t}{2}}X_N'\] where $Y_N$ is an $N\times N$ deterministic Hermitian matrix whose eigenvalue distribution converges as $N\to\infty$ and $X_N$ and $X_N'$ are independent Gaussian unitary ensembles. We also study various asymptotic behaviors of this Brown measure as the variance of the elliptic element approaches infinity.

math.OA↗

arXiv2022

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

math.OA↗

arXiv2022

The heat flow conjecture for polynomials and random matrices

We study the evolution of the roots of a polynomial of degree $N$, when the polynomial itself is evolving according to the heat flow. We propose a general conjecture for the large-$N$ limit of this evolution. Specifically, we propose (1) that the log potential of the limiting root distribution should evolve according to a certain first-order, nonlinear PDE, and (2) that the limiting root distribution at a general time should be the push-forward of the initial distribution under a certain explicit transport map. These results should hold for sufficiently small times, that is, until singularities begin to form. We offer three lines of reasoning in support of our conjecture. First, from a random matrix perspective, the conjecture is supported by a deformation theorem for the second moment of the characteristic polynomial of certain random matrix models. Second, from a dynamical systems perspective, the conjecture is supported by the computation of the second derivative of the roots with respect to time, which is formally small before singularities form. Third, from a PDE perspective, the conjecture is supported by the exact PDE\ satisfied by the log potential of the empirical root distribution of the polynomial, which formally converges to the desired PDE as $N\rightarrow \infty.$ We also present a "multiplicative" version of the the conjecture, supported by similar arguments. Finally, we verify rigorously that the conjectures hold at the level of the holomorphic moments.

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arXiv2022

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

We compute the Brown measure of $x_{0}+iσ_{t}$, where $σ_{t}$ is a free semicircular Brownian motion and $x_{0}$ is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region $Ω_{t}$ in the plane. In $Ω_{t},$ 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 $x_{0}+iσ_{t}$ by a certain map $Q_{t}:Ω_{t}\rightarrow\mathbb{R}$ gives the distribution of $x_{0}+σ_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+iσ_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.

math.PR↗

arXiv2022

Superconvergence in free probability limit theorems for arbitrary triangular arrays

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.

math.PR↗

arXiv2021

The Brown measure of unbounded variables with free semicircular imaginary part

Let $x_0$ be an unbounded self-adjoint operator such that the Brown measure of $x_0$ exists in the sense of Haagerup and Schultz. Also let $\tildeσ_α$ and $σ_β$ be semicircular variables with variances $α\geq 0$ and $β>0$ respectively. Suppose $x_0$, $σ_α$, and $\tildeσ_β$ are all freely independent. We compute the Brown measure of $x_0+\tildeσ_α+iσ_β$, extending the recent work which assume $x_0$ is a bounded self-adjoint random variable. We use the PDE method introduced by Driver, Hall and Kemp to compute the Brown measure. The computation of the PDE relies on a charaterization of the class of operators where the Brown measure exists. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+σ_{α+β}$. We also compute the example where $x_0$ is Cauchy-distributed.

math.OA↗

arXiv2021

Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix independent from $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$. For each $t>0$, the random matrix $A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is the circular variable of variance $t$, free from $x_0$. We use the Hamilton-Jacobi method to compute the Brown measure $ρ_t$ of $x_0+c_t$. The Brown measure has a density that is constant along the vertical direction inside the support. 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 semicircular variable of variance $t$, free from $x_0$. Furthermore, the push-forward of $ρ_t$ by a natural map is the law of $x_0+s_t$. Let $G_N(t)$ be the Brownian motion on the general linear group and let $U_N$ be a unitary random matrix independent from $G_N(t)$ such that $U_N$ converges in distribution to a unitary random variable $u$. The random matrix $U_NG_N(t)$ converges in $\ast$-distribution to $ub_t$ where $b_t$ is the free multiplicative Brownian motion, free from $u$. We compute the Brown measure $μ_t$ of $ub_t$, extending the recent work by Driver-Hall-Kemp, which corresponds to the case $u=I$. The measure has a density of the special form \[\frac{1}{r^2}w_t(θ)\] in polar coordinates in its support. The support of $μ_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion, free from $u$. The push-forward of $μ_t$ by a natural map is the law of $uu_t$. We compute the explicit formula for the special case where $u$ is Haar unitary. The support of the Brown measure of $ub_t$ is an annulus; in its support, the density in polar coordinates is given by \[\frac{1}{2πt}\frac{1}{r^2}.\]

math.OA↗

arXiv2020

A Local Limit Theorem and Delocalization of Eigenvectors for Polynomials in Two Matrices

We propose a boundary regularity condition for the $M_n(\mathbb{C})$-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove this through estimating the pair of $M_n(\mathbb{C})$-valued approximate subordination functions for the sum of two $M_n(\mathbb{C})$-valued random matrices $γ_1\otimes C_N+γ_2\otimes U_N^*D_NU_N$, where $C_N$, $D_N$ are deterministic diagonal matrices, and $U_N$ is Haar unitary.

math.PR↗

arXiv2017

Segal-Bargmann transform: the $q$-deformation

We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of Śniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.

math.PR↗

arXiv2016

The Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification

Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ limit of the two-parameter Segal-Bargmann transform on the unitary group $\mathbb{U}(N)$ and explore its limiting behavior. We also extend the notion of circular systems in order to define a two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin Theorem of the two-parameter free unitary Segal-Bargmann transform.

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