# UC San Diego Probability Seminar   2018-2019

#### Seminar Information

Thursdays, 10:30am   AP&M 6402
University of California, San Diego
La Jolla, CA 92093-0112
Map

Organizer:    Todd Kemp
E-mail:          tkemp@math.ucsd.edu

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2015-2016
2014-2015
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2011-2012
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2003-2004
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1999-2000
1998-1999

## Fall 2018 Schedule

Sept 27    Nikhil Srivastava, UC Berkeley
 Concentration for Sums of Random Matrices with Markov Dependence There are many well-known concentration results for sums of independent random matrices, e.g. those of Rudelson, Ahlswede-Winter, Tropp, and Oliveira. We move beyond the independent setting, and prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an reversible Markov chain, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which follows from complex interpolation methods. (Joint work with A. Garg, Y. Lee, and Z. Song.)

Oct 25    Li-Cheng Tsai, Columbia University
 Lower-tail large deviations of the KPZ equation Regarding time as a scaling parameter, we prove the one-point, lower tail Large Deviation Principle (LDP) of the KPZ equation, with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from [Borodin Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform.

Nov 8    Joshua Frisch, Caltech
 Proximal actions, Strong amenability, and infinite conjugacy class groups. A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points $p$ and $q$ we can simultaneously push them together i.e. there is a sequence $g_n$ such that $\lim g_n(p)=\lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $\mathbb{Z}$ acts proximally that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable.

Nov 15    Noah Forman, University of Washington
 The diffusion analogue to a tree-valued Markov chain In '99, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.

## "Winter" 2019 Schedule

Jan 10    Tom Kurtz, University of Wisconsin
 Population models as partial observations of genealogical models Classical models of biological populations, for example, Markov branching processes, typically model population size and possibly the distribution of types and/or locations of individuals in the population. The intuition behind these models usually includes ideas about the relationships among the individuals in the population that cannot be directly recovered from the model. This loss of information is even greater if one employs large population approximations such as the diffusion approximations popular in population genetics. "Lookdown" constructions provide representations of population models in terms of countable systems of particles in which each particle has a "type" which may record both spatial location and genetic type and a "level" which incorporates the lookdown structure which in turn captures the genealogy of the population. The original population model can then be viewed as the result of partial observation of the more complex model. We will exploit ideas from filtering of Markov processes to make the idea of partial observation clear and to justify the lookdown construction.

Jan 17    Yi Sun, Columbia University
 Gaussian fluctuations for products of random matrices This talk concerns singular values of $M$-fold products of i.i.d. right-unitarily invariant $N\times N$ random matrix ensembles. As $N\to\infty$, the height function of the Lyapunov exponents converges to a deterministic limit by work of Voiculescu and Nica-Speicher for $M$ fixed and by work of Newman and Isopi-Newman for $M$ tending to infinity with $N$. In this talk, I will show for a variety of ensembles that fluctuations of these height functions about their mean converge to explicit Gaussian fields which are log-correlated for $M$ fixed and have a white noise component for $M$ tending to infinity with $N$. These ensembles include rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values. I will sketch our technique, which derives a central limit theorem for global fluctuations via certain conditions on the multivariate Bessel generating function, a Laplace-transform-like object associated to the spectral measures of these matrix products. This is joint work with Vadim Gorin.

Jan 31    Haosui Duanmu, UC Berkeley
 Nonstandard analysis and its application to Markov processes Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in probability theory as well as stochastic processes. Nonstandard analysis allows construction of a single object - a hyperfinite probability space - which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly useful in porting discrete results into their continuous settings. In this talk, for every general-state-space continuous-time Markov process satisfying appropriate conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with each other up to some multiplicative constants for discrete-time general-state-space reversible Markov processes satisfying certain condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and Metropolis-Hasting algorithms.

Jan 31    Brian Hall, University of Notre Dame
Joint with Functional Analysis
 Eigenvalues of random matrices in the general linear groups I will consider random matrices in the general linear group GL(N;C) distributed according to a heat kernel measure. This may also be described as the distribution of Brownian motion in GL(N;C) starting at the identity. Numerically, the eigenvalues appear to cluster into a certain domain $\Sigma_t$ as $N$ tends to infinity. A natural candidate for the limiting eigenvalue distribution is the “Brown measure” of the limiting object, which is Biane’s "free multiplicative Brownian motion." I will describe recent work with Driver and Kemp in which we compute this Brown measure. The talk will be self contained and will have lots of pictures.

Feb 28    Hanbaek Lyu, UCLA
 Stable network observables via dynamic embedding of motifs We propose a novel framework for constructing and computing various stable network observables. Our approach is based on sampling a random homomorphism from a small motif of choice into a given network. Integrals of the law of the random homomorphism induces various network observables, which include well-known quantities such as homomorphism density and average clustering coefficient. We show that these network observables are stable with respect to renormalized cut distance between networks. For their efficient computation, we also propose two Markov chain Monte Carlo algorithms and analyze their convergence and mixing times. We demonstrate how our techniques can be applied to network data analysis, especially for hypothesis testing and hierarchical clustering, through analyzing both synthetic and real world network data.

March 7
APM 2402
Xin Sun, Columbia University
 Conformal embedding and percolation on the uniform triangulation Following Smirnov's proof of Cardy's formula and Schramm's discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice the has been achieved. Smirnorv's proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. Time permitting, I will also explain how this result fits in the the larger picture of random planar maps and Liouville quantum gravity.

## Spring 2019 Schedule

Apr 4
10:30am
Mark Meckes, Case Western University
 Quenched central limit theorem in a corner growth setting We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. The proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths. This is joint work with Christian Gromoll and Leonid Petrov.

Apr 4
1pm
Elizabeth Meckes, Case Western University
Department Colloquium
 Eigenvalues of random matrices: convergence of spectral measures and eigenvalue rigidity The behavior of the eigenvalues of large random matrices is generally very predictable, on multiple scales. Macroscopically, results like the semi-circle law describe the overall shape of the eigenvalue distributions. Indeed, for many natural ensembles of random matrices, we can describe in great detail the way the distribution of the eigenvalues converges to some limiting deterministic probability measure. On a microscopic scale, we often see the phenomenon of eigenvalue rigidity, in which individual eigenvalues concentrate strongly at predicted locations. I will describe some general approaches to these phenomena, with many examples: Wigner matrices, Wishart matrices, random unitary matrices, truncations of random unitary matrices, Brownian motion on the unitary group, and others.

Apr 11
4pm
Roman Vershynin, UC Irvine
Department Colloquium
 Mathematics of Deep Learning Deep learning is a rapidly developing area of machine learning, which uses artificial neural networks to perform learning tasks. Although mathematical description of neural networks is simple, theoretical explanation of spectacular performance of deep learning remains elusive. Even the most basic questions about it remain open. For example, how many different functions can a neural network compute? Jointly with Pierre Baldi (UCI CS) we discovered a general capacity formula for all fully connected boolean networks. The formula predicts, counterintuitively, that shallow networks have greater capacity than deep ones. So, mystery remains.

Apr 25
10:30am
Boguslaw Zegarlinski, Imperial College London
 Coercive inequalities for Markov generators on nilpotent Lie groups I will review and present some new results on construction, long time behaviour and coercive inequalities for Markov semigroups on nilpotent Lie groups.

May 30
10:30am
Mike Cranston, UC Irvine
 Some properties of the Riemann zeta distribution An alternative to selecting an integer uniformly from $1$ to $N$ and letting $N$ go to infinity is to select an integer according to the Riemann zeta distribution: the probability of selecting $n$ is $1/\zeta(s)n^s$, and letting $s$ go to $1$. We will explain several results that arise naturally due to the multiplicative property of this distribution.

June 6
10am
Larry Goldstein, USC
 Dickman Approximation in Quickselect sorting and Probabilistic Number Theory The generalized Dickman distribution ${\cal D}_\theta$ with parameter $\theta>0$ is the unique solution to the distributional equality $W=_d W^*$, where \begin{align*} W^*=_d U^{1/\theta}(W+1), \end{align*} with $W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of $W$, and $=_d$ denoting equality in distribution. Members of this family appear in the study of algorithms, number theory, stochastic geometry, and perpetuities. The Wasserstein distance $d(\cdot,\cdot)$ between such a $W$ with finite mean, and $D \sim {\cal D}_\theta$ obeys \begin{align*} d(W,D) \le (1+\theta)d(W^*,W). \end{align*} The specialization of this bound to the case $\theta=1$ and coupling constructions yield for $n \ge 1$ that \begin{align*} d_1(W_n,D) \le \frac{8\log (n/2)+10}{n} \quad \mbox{where } \quad W_n=\frac{1}{n}C_n-1, \end{align*} and $C_n$ is the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of $n$ distinct numbers. Joint with Bhattacharjee, using Stein's method, bounds for Wasserstein type distances can also be computed between ${\cal D}_\theta$ and weighted sums arising in probabilistic number theory of the form \begin{align*} S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k) \end{align*} where $(p_k)_{k \ge 1}$ is an enumeration of the prime numbers in increasing order and $X_k$ is, for instance, Geometric with parameter $1-1/p_k$.

June 6
11am
Amber Puha, Cal State San Marcos
Joint Stochastic Systems Seminar
 Asymptotic Behavior of a Critical Fluid Model for a Multiclass Processor Sharing Queue via Relative Entropy Queueing systems operating under the processor sharing discipline are relevant for studying time-sharing in computer and communication systems. Measure-valued processes, which track the residual service times of all jobs in the system, have been used to describe the dynamics of such systems. However, exact analysis of these infinite-dimensional stochastic processes is rarely possible. As a tool for approximate analysis of such systems, it has been proved that a fluid model arises as a functional law of large numbers limit of a multi-class processor sharing queue. This talk will focus on the asymptotic behavior of such a fluid model in the interesting regime of critical loading, where the average inflow of work to the system is equal to the capacity of the system to process that load. Using an approach involving a certain relative entropy functional, we show that critical fluid model solutions converge to a set of invariant states as time goes to infinity, uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time derivative of the relative entropy applied to trajectories of critical fluid model solutions. This is joint work with Justin A. Mulvany (USC) and Ruth J. Williams (UCSD).