# UC San Diego Probability Seminar   2018-2019

#### Seminar Information

Thursdays, 11am   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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## 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    Xin Sun, Columbia University
 TBA

## Spring 2019 Schedule

Apr 4    Elizabeth and Mark Meckes, Case Western University
 TBA

Apr 25    Boguslaw Zegarlinski, Imperial College London
 TBA