# Probability Seminar

# Spring 2017

**Thursdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.
**We usually end for questions at 3:15 PM.**

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.

## Monday, January 9, 4pm, B233 Van Vleck Miklos Racz, Microsoft Research

** Please note the unusual day and time **

Title: **Statistical inference in networks and genomics**

Abstract:
From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.

I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data.

## Thursday, January 26, Erik Bates, Stanford

Title: **The endpoint distribution of directed polymers**

Abstract: On the d-dimensional integer lattice, directed polymers are paths of a random walk in random environment, except that the environment updates at each time step. The result is a statistical mechanical system, whose qualitative behavior is governed by a temperature parameter and the law of the environment. Historically, the phase transitions have been best understood by whether or not the path’s endpoint localizes. While the endpoint is no longer a Markov process as in a random walk, its quenched distribution is. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods recently used by Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers. This talk is based on joint work with Sourav Chatterjee.

## Thursday, February 23, Jean-Luc Thiffeault, UW-Madison

**Title:** Heat Exchange and Exit Times

Abstract:

A heat exchanger can be modeled as a closed domain containing an incompressible fluid. The fluid has some temperature distribution obeying the advection-diffusion equation, with zero temperature boundary conditions at the walls. The goal is then to start from some initial positive heat distribution, and to flux it through the walls as fast as possible. Even for a steady flow, this is a time-dependent problem, which can be hard to optimize. Instead, we consider the mean exit time of Brownian particles starting from inside the domain. A flow favorable to heat exchange should lower the exit time, and so we minimize some norm of the exit time over incompressible flows (drifts) with a given energy. This is a simpler, time-independent optimization problem, which we then proceed to solve analytically in some limits, and numerically otherwise.

## Thursday, March 2, No Seminar this week

The talk by Thomas Woolley, Oxford has been moved to April 6 (see below).

## Thursday, March 16, Wei-Kuo Chen, Minnesota

Title: **Energy landscape of mean-field spin glasses**

Abstract:

The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloy, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of fruitful properties. This talk will be focused on the energy landscape of the SK model. First, we will present a formula for the maximal energy in Parisi’s formulation. Second, we will give a description of the energy landscape by showing that near any given energy level between zero and maximal energy, there exist exponentially many equidistant spin configurations. Based on joint works with Auffinger, Handschy, and Lerman.

## Thursday, March 23, Spring Break

## Wednesday, March 29, 1:00pm, Po-Ling Loh, UW-Madison

** Please note the unusual day and time
**

Title: **Confidence sets for the source of a diffusion in regular trees**

Abstract: We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential attachment and uniform attachment trees. At the core of our proofs is a probabilistic analysis of Polya urns corresponding to the number of uninfected neighbors in specific subtrees of the infection tree. We also describe extensions of our results to diffusions spreading over Galton-Watson trees. This is joint work with Justin Khim (UPenn).

## Thursday, April 6, Thomas Woolley, Oxford

Title: **Power spectra of stochastic reaction-diffusion equations on **
stochastically growing domains

Abstract: Being able to create and sustain robust, spatial-temporal
inhomogeneity is an important concept in developmental biology.
Generally, the mathematical treatments of these biological systems have
used continuum hypotheses of the reacting populations, which ignores any
sources of intrinsic stochastic effects. We address this concern by
developing analytical Fourier methods which allow us to probe the
probabilistic framework. Further, a novel description of domain growth
is produced, which is able to rigorously link the mean-field and stochastic
descriptions. Finally, through combining all of these ideas, it is shown
that the description of diffusion on a growing domain is non-unique and,
due to these distinct descriptions, diffusion is able to support
patterning without the addition of further kinetics.

## Thursday, 4/13/2017, Duncan Dauvergne, Toronto

Title: **The local limit of random sorting networks**

Abstract: A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of [math]S_n[/math] generated by adjacent transpositions. Remarkable conjectures about the global scaling limit of a uniformly random sorting network have been made based on strong empirical evidence. For example, trajectories of the individual elements 1, 2, … n appear to converge to sine curves.

One approach to proving these conjectures is to first show the existence of a local limit of random sorting networks, and then use this to piece together global information. In this talk, I will discuss this local limit, as well as progress that has been made towards understanding the global limit as a consequence of local properties.

## Thursday, April 20, Jinsu Kim, UW-Madison

Title : **Sufficient Conditions for Ergodicity of Stochastic Reaction Networks and Mixing Times**

Reaction networks are graphical configurations that can be used to describe biological interaction networks. If the abundances of the constituent species of the system are low, we can model the dynamics of species counts in a jump by jump fashion as a continuous time Markov chain. In this talk, we will mainly focus on which conditions of the graph imply ergodicity (existence of a stationary distribution) for the associated continuous time Markov chain. I will also present results related to their mixing times, which give the time required for the distribution of the continuous time Markov chain to get close to the stationary distribution.

## Wednesday, May 3, 1:00pm, Qin Li, UW-Madison

** Please note the unusual day and time
**

Title: **Stability of stationary inverse transport equation in diffusion scaling**

Abstract: We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE describes the dynamics of the distribution for photon particles. It often contains multiple scales characterized by the magnitude of a dimensionless parameter -- the Knudsen number (Kn). In the diffusive scaling (Kn << 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn \to 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Kn^p (p = 1 or 2), and as a result leads to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

## Thursday, May 11, 11:00 am Mihai Nica, Courant NYU

** Please note the unusual time
**

Title: **Intermediate disorder limits for multi-layer random polymers**

Abstract:

The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble, which I will also introduce. Part of this talk is based on joint work with Ivan Corwin.