UVA Probability Seminar
Wednesdays, 4:30 - 5:30pm
Kerchof 317
Organizers: Christian Gromoll, Tai Melcher, & Leonid Petrov
Mathematics Recent Seminars Other Seminars Maps & Directions

Fall 2014

3 Sep organizational meeting
24 Sep Leo Petrov* (joint with Algebra, meets at 3:30p)
Random tilings, representation theory and Macdonald processes I
1 Oct Leo Petrov* (joint with Algebra, meets at 3:30p)
Random tilings, representation theory and Macdonald processes II
3 Oct Leo Petrov* (joint with Algebra, meets 2-3p)
Random tilings, representation theory and Macdonald processes III
8 Oct Camelia Pop (U Penn)
Harnack inequalities for degenerate diffusions
29 Oct Xin Liu (Clemson)
Convergence of value functions for controlled subcritical stochastic networks
5 Nov Harry Crane (Rutgers)
Exchangeable combinatorial Markov processes
11 Nov Andrei Martinez-Finkelshtein* (University of Almeria, visiting Vanderbilt University); joint with Harmonic Analysis & PDE seminar, meets at 4pm
Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis
12 Nov Steve Pincus* (joint with Statistics, meets 4-5p in Clark 101)
Higher order dangers and precisely constructed taxa in models of randomness
18 Nov Ivan Corwin* (Columbia) meets 4-5p
Spectral theory and duality for interacting particle systems solvable by coordinate Bethe ansatz
3 Dec Rob Neel (Lehigh)
Degenerate martingales arising from various geometric structures, including minimal surfaces
* please note time, date, and/or location change

Abstracts

Random tilings, representation theory and Macdonald processes I

Leo Petrov* (joint with Algebra, meets at 3:30p)

I will discuss random lozenge tilings of polygons, their representation-theoretic interpretation (including connections to the character theory of the infinite-dimensional unitary group), and how this circle of ideas leads to Macdonald processes. The latter give rise to other probabilistic models such as random polymers and similar interacting stochastic particle systems
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Harnack inequalities for degenerate diffusions

Camelia Pop (U Penn)

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman-Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients, and on the a priori regularity of the weak solutions.
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Convergence of value functions for controlled subcritical stochastic networks

Xin Liu (Clemson)

In this talk, we study certain control problems for subcritical multiclass queueing networks. A multiclass queueing network $N$ consists of multiple classes of jobs and multiple servers. We assume that each class of jobs can only be served by one server, while each server could serve multiple classes of jobs. Such network is called subcritical if the traffic intensity at each server is not greater than 1. A server is called critical if its traffic intensity is equal to 1, and it is called strictly subcritical if its traffic intensity is strictly less than 1. Denote by $N_c$ the subnetwork consisting of the critical servers and the classes of jobs served by these servers. It is shown that the value function of the suitably scaled network control problem for the network $N$ converges to that of the diffusion control problem for a modified critical subnetwork that has the same dimension as $N_c$. Our result, in particular, shows how to reduce the dimension of the control problem for subcritical queueing networks, and also suggests a general strategy to construct near optimal controls.
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Exchangeable combinatorial Markov processes

Harry Crane (Rutgers)

We discuss some aspects of exchangeable Feller processes on combinatorial state spaces. These processes include many classical combinatorial stochastic processes, such as coalescent and fragmentation processes, as well as processes on more general state spaces. In the special case of partition- and graph-valued processes, we observe several nice properties, including a Lévy-Ito characterization of the jump rates. We also discuss some recent work that extends these properties to processes on general spaces of L-structures from first-order logic.
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Random Matrix Models, Non-intersecting random paths, and the Riemann-Hilbert Analysis

Andrei Martinez-Finkelshtein* (University of Almeria, visiting Vanderbilt University); joint with Harmonic Analysis & PDE seminar, meets at 4pm

Random matrix theory (RMT) is a very active area of research and a great source of exciting and challenging problems for specialists in many branches of analysis, spectral theory, probability and mathematical physics. The analysis of the eigenvalue distribution of many random matrix ensembles leads naturally to the concepts of determinantal point processes and to their particular case, biorthogonal ensembles, when the main object to study, the correlation kernel, can be written explicitly in terms of two sequences of mutually orthogonal functions.

Another source of determinantal point processes is a class of stochastic models of particles following non-intersecting paths. In fact, the connection of these models with the RMT is very tight: the eigenvalues of the so-called Gaussian Unitary Ensemble (GUE) and the distribution of random particles performing a Brownian motion, departing and ending at the origin under condition that their paths never collide are, roughly speaking, statistically identical.

A great challenge is the description of the detailed asymptotics of these processes when the size of the matrices (or the number of particles) grows infinitely large. This is needed, for instance, for verification of different forms of "universality" in the behavior of these models. One of the rapidly developing tools, based on the matrix Riemann-Hilbert characterization of the correlation kernel, is the associated non-commutative steepest descent analysis of Deift and Zhou.

Without going into technical details, some ideas behind this technique will be illustrated in the case of a model of squared Bessel nonintersecting paths.
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Higher order dangers and precisely constructed taxa in models of randomness

Steve Pincus* (joint with Statistics, meets 4-5p in Clark 101)

The certification, construction and delineation of individual, infinite length `random` sequences has been a longstanding, yet incompletely resolved problem. We address this topic via the study of normal numbers, which have often been viewed as reasonable proxies for randomness, given their limiting equidistribution of subblocks of all lengths. However, limitations arise within this perspective. First, we explicitly construct a normal number that satisfies the Law of the Iterated Logarithm (LIL), yet exhibits pairwise bias towards repeated values, rendering it inappropriate for any collection of random numbers. Accordingly, we deduce that the evaluation of higher order block dynamics, even beyond limiting equidistribution and fluctuational typicality, is imperative in proper evaluation of sequential `randomness`. Second, we develop several criteria motivated by classical theorems for symmetric random walks, which lead to algorithms for generating normal numbers that satisfy a variety of attributes for the series of initial partial sums, including rates of sign changes, patterns of return times to 0, and the extent of fairness of the sequence. Such characteristics are generally unaddressed in most evaluations of `randomness`. More broadly, we can differentiate normal numbers both on the basis of multiple distinct qualitative attributes, as well as quantitatively via a spectrum of rates within each attribute. Furthermore, we exhibit a toolkit of techniques to construct normal sequences that realize diverse a priori specifications, including profound biases. Overall, we elucidate the vast diversity within the category of normal sequences.
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Spectral theory and duality for interacting particle systems solvable by coordinate Bethe ansatz

Ivan Corwin* (Columbia) meets 4-5p

We describe recent advances in Bethe ansatz and Markov dualities related to the q-Hahn stochastic particle system. Limits of the system or its spectral theory have applications in studying various systems such as ASEP, q-TASEP, XXZ spin chain, delta Bose gas, and the KPZ equation. This is based off of joint work with Alexei Borodin, Leonid Petrov, and Tomohiro Sasamoto.
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Degenerate martingales arising from various geometric structures, including minimal surfaces

Rob Neel (Lehigh)

We first discuss a class of degenerate martingales (which we will call rank-n martingales) that arises naturally as the diffusion associated with minimal submanifolds, mean curvature flow, and some sub-Riemannian structures. This provides a unified approach to "coarse" properties, such as transience, of such structures. We then specialize to minimal surfaces in R^3, in which case the associated rank-2 martingale (which is just Brownian motion on the surface, viewed as a process in R^3) has the additional property that the tangent plane also evolves as a martingale. Taking advantage of this extra structure, we develop an extrinsic analogue of the mirror coupling of two Brownian motions. This allows us to study finer geometric and analytic properties of minimal surfaces, such as intersection results (strong halfspace-type theorems) and Liouville properties.
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