# Past Probability Seminars Fall 2009

# UW Math Probability Seminar Fall 2009

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by Benedek Valk�

## Schedule and Abstracts

|| Thursday, September 10 || || * David Anderson, * University of Wisconsin - Madison || || * Error analysis of numerical methods for stochastically modeled chemical reaction systems* ||

I will demonstrate an error analysis for numerical approximation methods for the generation of paths of continuous time Markov chain models commonly found in the chemistry, biochemistry, and biology literature. The motivation for the analysis is to be able to compare the accuracy of approximation methods and, specifically, standard tau-leaping and a midpoint method. I will demonstrate that the midpoint method achieves a higher order of accuracy, in both a weak and a strong sense, than standard tau-leaping; this result is in contrast to previous analyses and I will explain why.

|| Tuesday, September 15, 2:25 PM Van Vleck B325 || || * Alison Etheridge, * University of Oxford || || * Drift, draft and structure: modelling evolution in a spatial continuum* ||

One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating individuals in a sample of neutral genes from a panmictic population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees before describing a new approach (joint work with Nick Barton, IST Austria) to modelling the evolution of populations distributed in a spatial continuum.

|| Thursday, October 1 || || * B�lint Vir�g, * University of Toronto || || * Random matrix eigenvalues and Gaussian analytic functions * ||

There are two natural ways of producing repelling random point-clouds on the complex plane: either by taking the eigenvalues of some random matrix or by considering the zeros of a random (Gaussian) analytic function. I will explain some old and new connections between the two worlds.

|| Thursday, October 15 || || * No seminar because of* || || * MIDWEST PROBABILITY COLLOQUIUM* ||

|| Thursday, October 22 || || * Ivan Corwin, * Courant Institute || || * Directed last passage percolation with step-function boundary conditions* ||

Pr�hofer and Spohn (2002) conjectured a complete description of the fluctuations of a family of last passage percolation models with constant two-sided boundary conditions. Results of Baik, Ben Arous and P�ch� from Random Matrix Theory solve this problem for one-sided boundary conditions. Without appealing to any hard analysis, we use coupling techniques to prove the conjectured description for two-sided boundary conditions. We then consider last passage percolation with general step-function boundary conditions and show that these same coupling techniques lead to a complete description of the fluctuations in this case as well. The connection between this and TASEP (Totally Asymmetric Simple Exclusion Process) with general initial conditions is then pondered.

|| Thursday, October 29 || || * Lee DeVille, * University of Illinois at Urbana-Champaign || || * Synchrony-breaking and Rare Events in Stochastic Neuronal Networks* ||

We consider a model for a network of pulse-coupled oscillators containing randomness both in input and in network architecture. We analyze the scalings which arise in certain limits and interpret various "finite-size" effects as perturbations of these limits. Most notably, for certain parameters this network supports both synchronous and asynchronous modes of behavior and will switch stochastically between these modes due to rare events. We also relate the analysis of this network to classical results in graph theory, and in particular, those involving the size of the "giant component" in the Erdos-Renyi random graph. This work is joint with Charles Peskin and Joel Spencer.

|| Thursday, November 5, 4PM, Van Vleck B102 || || * Persi Diaconis, * Stanford || || * HILDALE LECTURE SERIES* ||

|| Thursday, November 12 || || * Fedja Nazarov *, University of Wisconsin - Madison || || * Abundance of maximal paths in the random grid.* ||

We show that if the vertices of $Z^d$ are given independent weights $0$ and $1$ with probability $p$ and $1-p$ respectively, then the number of diagonal paths joining the origin and the point $(n,\dots n)$ that contain the maximal possible number of $1$'s is exponentially large in $n$ with probability close to $1$. (here by a diagonal path we mean a path whose each step increases exactly one coordinate by $1$). This is a joint work with Yuval Peres and Vladas Sidoravicius.

|| Thursday, November 19 || || * Ton Dieker, * Georgia Institute of Technology || || * Interlacings and the interchange process on weighted graphs* ||

A central question in the theory of card shuffling is how quickly a deck of cards becomes 'well-shuffled' given a shuffling rule. In this talk, I will discuss a probabilistic card shuffling model known as the 'interchange process'. A 1992 conjecture by Aldous and Diaconis about this model has recently been resolved (see http://www.stat.berkeley.edu/~aldous/Research/OP/sgap.html ) and I will discuss how my work has been involved with this.

|| Monday, November 23, 2:25 PM Van Vleck B135 || || * Uwe Einmahl, * Vrije Universiteit Brussel || || * Strong invariance principles and a generalization of the law of the iterated logarithm* ||

One of the classical results of probability is the Hartman-Wintner (1941) law of the iterated logarithm (LIL) which allows us to determine the precise order of magnitude of sums of i. i. d. mean zero random variables with finite variance. This result was later improved to a functional LIL by Strassen (1964). The crucial tool for proving this result was a strong approximation of sums of i. i. .d. random variables X_1, X_2, .... by a suitable Brownian motion. After the seminal work by Strassen (1964) the question was raised whether strong approximations with better rates are possible if one assumes that higher moments exist or if X_1 even has a finite moment generating function. This problem was eventually solved by Komlos, Major and Tusnady (1976). After the work of these authors the question still remained open of what are the best possible strong approximation rates if one has no higher moments, but if one knows, for instance, that E[X^2 log (1 +|X|)] is finite. This problem in turn was solved by Einmahl (1987). Later Einmahl and Mason (1993) suggested a slightly different form of strong invariance principle which implies the results of Einmahl (1987) and a part of the results of Komlos, Major, Tusnday (1976) among other things. In this talk I will present a recent extension of the strong invariance principle of Einmahl and Mason (1993) to the infinite variance case and I will show how one can infer generalizations of the Hartman-Wintner LIL along with suitable functional LIL type results from this generalized strong invariance principle. Our proof is based on a new strong approximation technique which is due to Sakhanenko (2000) and which works also for d-dimensional random vectors. This enables us to prove most of our results for d-dimensional random vectors.

|| Thursday, December 3 || || * David Griffeath, * University of Wisconsin - Madison || || * Asymptotic densities for Packard Snowflakes with box neighborhood * ||

In 1984, Norman Packard introduced a family of very simple deterministic, two-dimensional lattice growth models that generate crystals reminiscent of the variety of snowflakes known as stellar plates. On the diamond and triangle lattices all cases can be understood rigorously. I will discuss the much more delicate box case, where each cell has eight nearest neighbors, focusing on the asymptotic density of the final configuration. When sites with 1 or 2 occupied neighbors solidify, additivity yields a comprehensive result for general finite initial sets. In particular, the density is independent of the seed. In the case that requires either 1 or 3 occupied neighbors for solidification, we show that different seeds may produce many different densities. More generally, analysis is restricted to special initializations, cases with density 1, and various experimental findings. For instance, if solidification requires exactly 1 occupied neighbor, then a snowflake pattern emerges from some seeds, but others fill the plane with a chaotic state. An attempt to understand this last case leads inevitably to the subject of my second talk.

Despite the lack of randomness in the dynamics, our methods are related to those used nearly 30 years ago to study stochastic particle systems such as Richardson's growth model. I will try to highlight these connections with probability theory. (Joint with Janko Gravner) For details, see: Nonlinearity 22 (2009) 1817-1846.

|| Thursday, December 10 || || * David Griffeath, * University of Wisconsin - Madison || || * The one-dimensional Exactly 1 cellular automaton: replication, periodicity, and chaos from finite seeds * ||

In the Exactly 1 cellular automaton, every site of the one-dimensional lattice is either in state 0 or in state 1, and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three-site neighborhood at the current time. I will explore this rule started from finite seeds, i.e., those initial configurations that have only finitely many 1's. Three qualitatively different types of evolution are observed: replication, periodicity, and chaos. I will focus on rigorous results, assisted by algorithmic searches, for the first two behaviors. In particular, I'll explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds. Some empirical observations about chaotic seeds will also be presented.

Exactly 1 arises as the edge dynamics for some of the Packard snowflakes described in the first talk. So I will show how understanding of the one-dimensional rule has implications for the two-dimensional growth. As a puzzle, try to find an initial seed that generates a space-time periodic pattern. In the space-time diagram for such a seed (with time running down as is customary), each occupied site after the top row should have exactly one occupied neighbor to its NW, N, or NE in the previous row, and a periodic tiling should emerge as the growth spreads. The smallest known seed of this type has length 83. (Joint with Janko Gravner) For details on this very recent work, see: http://psoup.math.wisc.edu/exactly1/exactly1.html