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Brian C. Hall

Publications and source records attributed to Brian C. Hall.

At least 19 recordsLinked to original sources
arXiv2026

Repeated differentiation of deterministic polynomials with asymptotically radial root distributions

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P(z)=p(z^m)$, where $p$ is a deterministic polynomial of degree $n$ with real, non-negative roots, in the regime where $m$ and $n$ are large. If $m\gg \log(n)$ and the root distribution of $P$ converges to a compactly supported, radial probability measure $\mu_0$, these works show that for $0\le t<1$, the root distribution of the $\lfloor nmt\rfloor$-th derivative of $P$ converges to a compactly supported probability measure $\mu_t$ given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator $z^a(d/dz)^b$. We also compute the limiting root distribution in the case when $m$ is fixed and $n$ tends to infinity.

math.PR↗
arXiv2025

Support of the Brown measure of a family of free multiplicative Brownian motions with non-negative initial condition

We consider a family $b_{s,τ}$ of free multiplicative Brownian motions labeled by a real variance parameter $s$ and a complex covariance parameter $τ$. We then consider the element $xb_{s,τ}$, where $x$ is non-negative and freely independent of $b_{s,τ}$. Our goal is to identify the support of the Brown measure of $xb_{s,τ}$. In the case $τ=s$, we identify a region $Σ_s$ such that the Brown measure is vanishing outside of $\overlineΣ_s$ except possibly at the origin. For general values of $τ$, we construct a map $f_{s-τ}$ and define $D_{s,τ}$ as the complement of $f_{s-τ}(\overlineΣ_s^c)$. Then the Brown measure is zero outside $D_{s,τ}$ except possibly at the origin. The proof of these results is based on a two-stage PDE analysis, using one PDE (following the work of Driver, Hall, and Kemp) for the case $τ=s$ and a different PDE (following the work of Hall and Ho) to deform the $τ=s$ case to general values of $τ$.

math.PR↗
arXiv2025

Matrix Random Walks and the Lima Bean Law

A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to Brownian motion $B(t)$ on a matrix Lie group. In this paper, we study the eigenvalues of such rescaled matrix random walks, as $N\to\infty$ and $k\to\infty$. The standard Brownian motion $W(t)$ on $\mathrm{M}_N(\mathbb{C})$ has independent Gaussian entries at each $t$. It is bi-invariant: mutiplying on the left or right by a unitary does not change the distribution. We prove that the empirical eigenvalue distribution of any matrix random walk $B_k$ with bi-invariant steps $A_j$ and initial distribution converges (for fixed $k$ as $N\to\infty$) to a probability measure on $\mathbb{C}$: the Brown measure of the free probability $\ast$-distribution limit $b_k$ of the random walk. If the steps $A_j$ are identically distributed with normalized Hilbert--Schmidt norm $\|A_j\|_2 = t$, the limit law of eigenvalues is supported on a compact ``lima bean'' shaped region. We explicitly compute the limit measure and region, and characterize their phase transitions as $t$ evolves. We prove that the Brown measure of $b_k$ converges as $k\to\infty$, to the Brown measure of the free multiplicative Brownian motion, assuming only that the steps are bi-invariant and normalized in Hilbert--Schmidt norm. Thus the Brownian motion is the universal limit of rescaled matrix random walks, under very general assumptions on the distribution of steps.

math.PR↗
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

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.

math.PR↗
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.

math.PR↗
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.

math.PR↗
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↗
arXiv2020

PDE methods in random matrix theory

This article begins with a brief review of random matrix theory, followed by a discussion of how the large-$N$ limit of random matrix models can be realized using operator algebras. I then explain the notion of "Brown measure," which play the role of the eigenvalue distribution for operators in an operator algebra. I then show how methods of partial differential equations can be used to compute Brown measures. I consider in detail the case of the circular law and then discuss more briefly the case of the free multiplicative Brownian motion, which was worked out recently by the author with Driver and Kemp.

math.PR↗
arXiv2020

The Brown measure of the free multiplicative Brownian motion

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 $Σ_{t}$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_{t}$ on $\barΣ_{t},$ which is strictly positive and real analytic on $Σ_{t}$. This density has a simple form in polar coordinates: \[ W_{t}(r,θ)=\frac{1}{r^{2}}w_{t}(θ), \] where $w_{t}$ is an analytic function determined by the geometry of the region $Σ_{t}$. 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 Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

math.PR↗
arXiv2018

The large-N limit for two-dimensional Yang-Mills theory

The analysis of the large-$N$ limit of $U(N)$ Yang-Mills theory on a surface proceeds in two stages: the analysis of the Wilson loop functional for a simple closed curve and the reduction of more general loops to a simple closed curve. In the case of the 2-sphere, the first stage has been treated rigorously in recent work of Dahlqvist and Norris, which shows that the large-$N$ limit of the Wilson loop functional for a simple closed curve in $S^{2}$ exists and that the associated variance goes to zero. We give a rigorous treatment of the second stage of analysis in the case of the 2-sphere. Dahlqvist and Norris independently performed such an analysis, using a similar but not identical method. Specifically, we establish the existence of the limit and the vanishing of the variance for arbitrary loops with (a finite number of) simple crossings. The proof is based on the Makeenko-Migdal equation for the Yang-Mills measure on surfaces, as established rigorously by Driver, Gabriel, Hall, and Kemp, together with an explicit procedure for reducing a general loop in $S^{2}$ to a simple closed curve. The methods used here also give a new proof of these results in the plane case, as a variant of the methods used by Lévy. We also consider loops on an arbitrary surface $Σ$. We put forth two natural conjectures about the behavior of Wilson loop functionals for topologically trivial simple closed curves in $Σ.$ Under the weaker of the conjectures, we establish the existence of the limit and the vanishing of the variance for topologically trivial loops with simple crossings that satisfy a "smallness" assumption. Under the stronger of the conjectures, we establish the same result without the smallness assumption.

hep-th↗
arXiv2018

A unitary "quantization commutes with reduction" map for the adjoint action of a compact Lie group

Let $K$ be a simply connected compact Lie group and $T^{\ast}(K)$ its cotangent bundle. We consider the problem of "quantization commutes with reduction" for the adjoint action of $K$ on $T^{\ast}(K).$ We quantize both $T^{\ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{\ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.

math-ph↗
arXiv2017

Coherent states for compact Lie groups and their large-N limits

The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform for the unitary group U(N). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."

math-ph↗
arXiv2017

The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces

We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

math-ph↗
arXiv2016

Three proofs of the Makeenko-Migdal equation for Yang-Mills theory on the plane

We give three short proofs of the Makeenko-Migdal equation for the Yang-Mills measure on the plane, two using the edge variables and one using the loop or lasso variables. Our proofs are significantly simpler than the earlier pioneering rigorous proofs given by T. Lévy and by A. Dahlqvist. In particular, our proofs are "local" in nature, in that they involve only derivatives with respect to variables adjacent to the crossing in question. In an accompanying paper with F. Gabriel, we will show that two of our proofs can be adapted to the case of Yang-Mills theory on a compact surface.

math-ph↗
arXiv2013

The Segal-Bargmann transform for unitary groups in the large-N limit

This paper describes results of the author with B. K. Driver and T. Kemp concerning the large-N limit of the Segal--Bargmann transform for the unitary group U(N). We consider the transform on matrix-valued functions that are polynomials in a single variable in U(N). We show that in the large-N limit, the transform maps functions of this type to single-variable polynomial functions on the complex group GL(N;C). This result was conjectured by Ph. Biane and was also proved independently by G. Cébron. The first main ingredient in our proof of this result is an "asymptotic product rule" for the Laplacian on U(N), which allows us to compute explicitly the leading-order large-N behavior of the heat operator on U(N). The second main ingredient in the proof is the phenomenon of "concentration of traces," in which the relevant heat kernel measures are concentrating onto sets where the trace of any power of the variable is constant.

math.RT↗