About

Todd Kemp received his Ph.D. in Mathematics from Cornell University in 2005. He held an Escobar Assistant Professorship at Cornell during 2005-2006 and was a CLE Moore Instructor at MIT from 2006-2009 before joining the faculty at UCSD in 2009. Kemp's research is in Probability Theory and Functional Analysis, with particular focus on random matrices, free probability theory, heat kernel analysis, and functional inequalities.

Research topics

• Artificial Intelligence
• Computer Science
• Mathematics
• Algorithm
• Combinatorics
• Philosophy
• Statistics

Selected publications

• Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

  ArXiv.org · 2025-11-13

  preprintOpen access

  We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{ρ,ζ}\colon |ζ|<ρ\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{ρ,ζ}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $δ>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq δ\right)\leq \left(C t/δ\right)^{N^{2/3}} \] for a constant $C=C_ρ$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

  PublisherOA PDFDOI

• The Strong Haagerup Inequality for <i>q</i> -Circular Systems

  International Mathematics Research Notices · 2025-11-27

  article1st authorCorresponding

  Abstract Together with Speicher, in 2007, the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with $\sqrt{n}$, rather than linearly in $n$, on homogeneous polynomials of degree $n$. In this paper, we prove a similar inequality for $q$-circular systems for $|q|&amp;lt;1$, generalizing the free case when $q=0$. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor $\mathscr{R}$-diagonality. As an application, we prove a strong ultracontractivity theorem for the $q$-Ornstein–Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.

  PublisherDOI

• A martingale approach to noncommutative stochastic calculus

  Journal of Functional Analysis · 2025-09-19

  articleOpen access

  We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the q-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder–Davis–Gundy inequalities (with p ≥ 2 [jls-end-space/]) for continuous-time noncommutative martingales and a noncommutative Itô's formula for “adapted C²maps,” including trace ⁎-polynomial maps and operator functions associated to the noncommutative C²scalar functions R → C introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative C²functions introduced by Jekel, Li, and Shlyakhtenko.

  PublisherDOI

• Matrix Random Walks and the Lima Bean Law

  ArXiv.org · 2025-10-12

  preprintOpen access

  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.

  PublisherOA PDFDOI

• Double $q$-Wigner Chaos and the Fourth Moment

  ArXiv.org · 2025-11-25

  preprintOpen access1st authorCorresponding

  In this paper, we prove the Fourth Moment Theorem for sequences of (noncommutative) random variables given as sums of two stochastic integrals in two different parity orders of chaos, both in the free Wigner chaos setting and a $q$-Gaussian generalization. Specifically, we prove that convergence to the appropriate central limit distribution is mediated entirely by the behavior of the first four (mixed) moments of the two stochastic integrals, which in turn controls the $L^2$ norms of partial integral contractions of those kernels. The key step in both the free and $q$-Gaussian settings is a polarization identity for fourth cumulants of sums which holds only when the two terms have differing parities. These results are analogous to the recent preprint Fourth-Moment Theorems for Sums of Multiple Integrals by Basse-O'Connor, Kramer-Bang, and Svedsen in the classical Wiener-Itô chaos setting.

  PublisherOA PDFDOI

• The strong Haagerup inequality for q-circular systems

  arXiv (Cornell University) · 2024-09-05

  preprintOpen access1st authorCorresponding

  Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with $\sqrt{n}$, rather than linearly in $n$, on homogeneous polynomials of degree $n$. In this paper, we prove a similar inequality for $q$-circular systems for $|q|&lt;1$, generalizing the free case when $q=0$. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor $\mathscr{R}$-diagonality. As an application, we prove a strong ultracontractivity theorem for the $q$-Ornstein--Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.

  PublisherOA PDFDOI

• Bias and Division in the Free World

  arXiv (Cornell University) · 2024-03-28

  preprintOpen accessSenior author

  Sampling bias is a foundational concept in statistics; associated bias transforms, such as size bias, have come to play important roles in probability theory of late. The first author and G. Reinert introduced zero bias, a transform whose unique fixed point is the normal distribution; it has become a standard tool in Stein's method and Gaussian approximation. Very recently, connections between zero bias and the class of infinitely divisible distributions have been found. In this paper, we develop a free probabilistic analog of the zero bias transform, proving its existence and regularity. The free zero bias has the semicircle law (free probability's central limit distribution) as its unique fixed point. We offer a construction of the free zero bias that mirrors a classical one incorporating square bias with a mollifier, and in the process develop a surprisingly new class of distributional operations through their Cauchy transforms. We then explore connections between the free zero bias, and size bias, with the class of freely infinitely divisible distributions. We develop a new self-contained treatment of the subject, together with a new characterization of free infinite divisibility using bias transforms. We also develop a parallel treatment of positively freely infinitely divisible distributions, which can also be characterized by a new kind of Levy--Khintchine formula that has no known classical analogue, and we use this to both give several new descriptions of such distributions and furnish new examples using these bias methods.

  PublisherOA PDFDOI

• A martingale approach to noncommutative stochastic calculus

  arXiv (Cornell University) · 2023-08-18

  preprintOpen access

  We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the $q$-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder--Davis--Gundy inequalities (with $p \geq 2$) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted $C^2$ maps," including trace $\ast$-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $\mathbb{R} \to \mathbb{C}$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko.

  PublisherOA PDFDOI

• Fluctuations of Brownian motions on GLN

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2022-02-01 · 4 citations

  preprintOpen accessSenior author

  Nous considérons une famille à deux paramètres de processus unitairement invariants sur le groupe général linéaire GLN des matrices N×N inversibles, contenant comme cas particuliers le mouvement brownien standard ainsi que le mouvement brownien unitaire. Nous montrons que tous ces processus ont des fluctuations spectrales gaussiennes d’ordre O(1N) en grande dimension ; ces fluctuations sont établies pour les distributions finies-dimensionnelles du processus sous une classe étendue de fonctions tests appelées polynômes à trace. Nous donnons une expression explicite de la covariance du champ gaussien des fluctuations en fonction d’une fonctionnelle particulière de trois mouvements browniens multiplicatifs librement indépendants. Ces résultats généralisent les précédents travaux de Lévy et Maïda, et de Diaconis et Evans, sur les groupes unitaires.

  PublisherOA PDFDOI

• The Brown measure of the free multiplicative Brownian motion

  Probability Theory and Related Fields · 2022 · 14 citations

  Senior authorCorresponding
  • Artificial Intelligence
  • Computer Science
  • Algorithm

  Abstract The free multiplicative Brownian motion $$b_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> is the large- N limit of the Brownian motion on $$\mathsf {GL}(N;\mathbb {C}),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>;</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> in the sense of $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -distributions. The natural candidate for the large- N limit of the empirical distribution of eigenvalues is thus the Brown measure of $$b_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $$\Sigma _{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $$W_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> on $$\overline{\Sigma }_{t},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mover> <mml:mi>Σ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which is strictly positive and real analytic on $$\Sigma _{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> . This density has a simple form in polar coordinates: $$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfrac> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>θ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$w_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> is an analytic function determined by the geometry of the region $$\Sigma _{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Σ</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> . We show also that the spectral measure of free unitary Brownian motion $$u_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> is a “shadow” of the Brown measure of $$b_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

  PublisherOA PDFDOI

Recent grants

• Functional Inequalities in Global Analysis and Non-Communitative Geometry

  NSF · $117k · 2007–2010

• CAREER: Free Probability and Connections to Random Matrices, Stochastic Analysis, and PDEs

  NSF · $550k · 2013–2019

• Stochastic Differential Equations, Heat Kernel Analysis, and Random Matrix Theory

  NSF · $210k · 2018–2023

• RANDOM MATRICES IN FUNCTIONAL ANALYSIS

  NSF · $124k · 2010–2013

Frequent coauthors

• Brian C. Hall

  12 shared

• Ioana Dumitriu

  University of California, San Diego

  9 shared

• Kavita Ramanan

  9 shared

• Bruce K. Driver

  University of California, San Diego

  9 shared

• Jean-Jacques Lœb

  7 shared

• Piotr Graczyk

  7 shared

• Roland Speicher

  5 shared

• C. Benoît

  Kyoto University

  5 shared

Awards & honors

• Academic Senate Distinguished Teaching Award
• NSF CAREER Award
• Hellman Fellowship