# Past Probability Seminars Fall 2016

# Fall 2016

## Thursday, September 8, Daniele Cappelletti, UW-Madison

Title: **Reaction networks: comparison between deterministic and stochastic models**

Abstract: Mathematical models for chemical reaction networks are widely used in biochemistry, as well as in other fields. The original aim of the models is to predict the dynamics of a collection of reactants that undergo chemical transformations. There exist two standard modeling regimes: a deterministic and a stochastic one. These regimes are chosen case by case in accordance to what is believed to be more appropriate. It is natural to wonder whether the dynamics of the two different models are linked, and whether properties of one model can shed light on the behavior of the other one. Some connections between the two modelling regimes have been known for forty years, and new ones have been pointed out recently. However, many open questions remain, and the issue is still largely unexplored.

## Friday, September 16, 11 am Elena Kosygina, Baruch College and the CUNY Graduate Center

** Please note the unusual day and time **

The talk will be in Van Vleck 910 as usual.

Title: **Homogenization of viscous Hamilton-Jacobi equations: a remark and an application.**

Abstract: It has been pointed out in the seminal work of P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan that for the first order Hamilton-Jacobi (HJ) equation, homogenization starting with affine initial data should imply homogenization for general uniformly continuous initial data. The argument utilized the properties of the HJ semi-group, in particular, the finite speed of propagation. The last property is lost for viscous HJ equations. We remark that the above mentioned implication holds under natural conditions for both viscous and non-viscous Hamilton-Jacobi equations. As an application of our result, we show homogenization in a stationary ergodic setting for a special class of viscous HJ equations with a non-convex Hamiltonian in one space dimension. This is a joint work with Andrea Davini, Sapienza Università di Roma.

## Thursday, September 22, Philip Matchett Wood, UW-Madison

Title: **Low-degree factors of random polynomials**

Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers. It is known that certain models are very likely to produce random polynomials that are irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in fact, a universal behavior. For example, we show that the characteristic polynomial of random matrix with independent +1 or −1 entries is very unlikely to have a factor of degree up to [math]\displaystyle{ n^{1/2-\epsilon} }[/math]. Joint work with Sean O’Rourke. The talk will also discuss joint work with UW-Madison undergraduates Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg, who were supported by NSF grant DMS-1301690 and co-supervised by Melanie Matchett Wood.

## Thursday, September 29, Joseph Najnudel, University of Cincinnati

Title: **On the maximum of the characteristic polynomial of the Circular Beta Ensemble**

In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.

## Thursday, October 6, No Seminar

## Thursday, October 13, No Seminar due to Midwest Probability Colloquium

For details, see Midwest Probability Colloquium.

## Thursday, October 20, Amol Aggarwal, Harvard

Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model

Abstract: We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.

## Thursday, October 27, Hung Tran, UW-Madison

Title: **Homogenization of non-convex Hamilton-Jacobi equations**

Abstract: I will describe why it is hard to do homogenization for non-convex Hamilton-Jacobi equations and explain some recent results in this direction. I will also make a very brief connection to first passage percolation and address some challenging questions which appear in both directions. This is based on joint work with Qian and Yu.

## Thursday, November 3, Alejandro deAcosta, Case-Western Reserve

Title: **Large deviations for irreducible Markov chains with general state space**

Abstract: We study the large deviation principle for the empirical measure of general irreducible Markov chains in the tao topology for a broad class of initial distributions. The roles of several rate functions, including the rate function based on the convergence parameter of the transform kernel and the Donsker-Varadhan rate function, are clarified.

## Thursday, November 10, Louis Fan, UW-Madison

Title: **Particle representations for (stochastic) reaction-diffusion equations**

Abstract:

Reaction diffusion equations (RDE) is a popular tool to model complex spatial-temporal patterns including Turing patterns, traveling waves and periodic switching.

These models, however, ignore the stochasticity and individuality of many complex systems in nature. Recognizing this drawback, scientists are developing individual-based models for model selection purposes. The latter models are sometimes studied under the framework of interacting particle systems (IPS) by mathematicians, who prove scaling limit theorems to connect various IPS with RDE across scales.

In this talk, I will present some new limiting objects including SPDE on metric graphs and coupled SPDE. These SPDE reduce to RDE when the noise parameter tends to zero, therefore interpolates between IPS and RDE and identifies the source of stochasticity. Scaling limit theorems and novel duality formulas are obtained for these SPDE, which not only connect phenomena across scales, but also offer insights about the genealogies and time-asymptotic properties of certain population dynamics. In particular, I will present rigorous results about the lineage dynamics for of a biased voter model introduced by Hallatschek and Nelson (2007).

## Thursday, November 24, No Seminar due to Thanksgiving

## Thursday, December 1, Hao Shen, Columbia

Title: **On scaling limits of Open ASEP and Glauber dynamics of ferromagnetic models**

Abstract: We discuss two scaling limit results for discrete models converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP. We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. A common challenge in the proofs is how to identify the limiting process as the solution to the SPDE, and we will discuss how to overcome the difficulties in the two cases.(Based on joint works with Ivan Corwin and Hendrik Weber.)

**Colloquium** Friday, December 2, Hao Shen, Columbia

4pm, Van Vleck 9th floor

Title: **Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?**

Abstract: Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.