# Past Probability Seminars Spring 2005

## UW Math Probability Seminar Spring 2005

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

Organized by [index.html Timo Sepp�l�inen ]

### Schedule and Abstracts

|| Thursday, January 20 || || *David Revelle, * UC Berkeley || || * Scaling limits for the uniform spanning tree on the discrete torus * ||

The typical distance between two points in the uniform spanning tree (UST) on the complete graph K_m is on the order of m^{1/2}, and Aldous showed that a suitable scaling limit of the UST is the Brownian continuum random tree. We will show that for d\geq 5, the scaling limit of the UST on the discrete torus Z_n^d is again the Brownian continuum random tree. This verifies a conjecture of Pitman. In the talk we will describe the Brownian continuum random tree as well as explain Wilson's algorithm between the UST and loop-erased random walk. No previous familiarity with either of these topics will be assumed. This is joint work with Yuval Peres.

|| Thursday, January 27 || || * Sharad Goel, * Cornell University || || * Mixing times for top to bottom shuffles * ||

A deck of n cards is shuffled by repeatedly moving the top card to one of the bottom k_n positions of the deck uniformly at random. How many shuffles does it take to randomize the deck? For k_n = n, this is the well-studied top to random walk and n log n shuffles are required; for k_n = 2, this is the Rudvalis shuffle and order n^3 log n shuffles are needed. I plan to discuss the mixing time for this family of shuffles as k_n ranges from a constant to n, showing that for large k_n the walks behave like the top to random shuffle, while for small k_n the walks behave like the Rudvalis shuffle. Some of the tools used to analyze these walks include coupling, Wilson's lemma, and comparison techniques for random walks on groups.

|| Thursday, February 3 || || * Anastasia Ruzmaikina,* Purdue University || || * Characterization of invariant measures at the leading edge for competing particle systems * ||

We study systems of particles on a line which have a maximum, are locally finite, and evolve with independent increments. "Quasi-stationary states" are defined as probability measures, on the sigma-algebra generated by the gap variables, for which the joint distribution of the gaps is invariant under the time evolution. Examples are provided by Poisson processes with densities of the form, rho(dx)=e^{- sx}s dx, with s > 0, and linear superpositions of such measures. We show that conversely: any quasi-stationary state for the independent dynamics, with an exponentially bounded integrated density of particles, corresponds to a superposition of the above described probability measures, restricted to the relevant sigma-algebra. Among the systems for which this question is of some relevance are spin-glass models of statistical mechanics, where the point process represents the collection of the free energies of distinct "pure states", the time evolution corresponds to the addition of a spin variable, and the Poisson measures described above correspond to the so-called REM states.

|| Thursday, February 10 || || * * || || * * ||

|| Thursday, February 17 || || * Nathana�l Berestycki, * Cornell University || || * Phase transition and geometry of random transpositions * ||

We are originally motivated by a problem in genome rearrangement. One way to formulate this problem is to ask what is the rate of evolution induced by certain large-scale mutations called inversions. If we think of a chromosome as being made up of n markers (the genes), a reversal is a mutation that chooses a segment of the chromosome and flips it around to reverse its order. Traditionally, biologists have studied these questions with parsimony methods. However, it was observed numerically by Hannehalli and Pevzner that this method provides accurate results only if the number of mutations is small enough. Our work provides a theoretical explanation for this fact and allows to consider the case of many mutations. We consider the mathematically cleaner but similar problem of random transpositions. Let sigma(t) be the composition of random uniform transpositions performed at rate 1. sigma(t) can be viewed as a random walk on some Cayley graph of the symmetric group. If D(t) is the distance between sigma(t) and its starting point, we prove that D(t) undergoes a phase transition at critical time n/2, from a linear to a sublinear behavior. We will also describe the consequences of this result for the geometry of the graph, which involves a connection with Gromov hyperbolic spaces.

|| Thursday, February 24 || || * James Martin, * University of Paris 7 || || * Stationary distributions of multi-type totally asymmetric exclusion processes * ||

We consider totally asymmetric simple exclusion processes with n types of particle and holes (n-TASEPs). Omer Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angel's construction can be interpreted in terms of the operation of a discrete-time M/M/1 queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a system of queues in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose evolutions are coupled but whose distributions at any fixed time are independent. Joint work with Pablo Ferrari.

|| Thursday, March 3 || || * Tom Kurtz, * UW-Madison || || * Multiscale approximations for stochastic reaction networks * ||

A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic model for a network treats the system as a continuous time Markov chain whose state is a vector giving the number of molecules of each species present with each reaction modeled as a possible transition for the state. In classical chemistry, systems are so large that stochastic fluctuations are irrelevant and reaction networks are modeled with systems of ordinary differential equations. Interest in modeling chemical reactions within biological cells has led to renewed interest in stochastic models, since the number of molecules involved, at least for some of the species, may be sufficiently small that the deterministic model does not provide a good representation of the behavior of the system. Modeling is complicated by the fact that some species may be present in much greater abundance than others. In addition, the rate constants may vary over several orders of magnitude. With these two issues in mind, we consider approaches to the approximation of the stochastic models that take the multiscale nature of the system into account.

|| Thursday, March 10 || || * * || || * * ||

|| Thursday, March 17 || || * * || || * * ||

|| Thursday, March 24 || || SPRING BREAK ||

|| Thursday, March 31 || || * * || || * * ||

|| Thursday, April 7 || || * Stanislav Volkov, * University of Bristol || || * Review of reinforced random walks and processes * ||

Reinforced processes are stochastic processes (discrete or continuous time) on graphs which exhibit "homesick" behavior, namely the like visit the "places" the have already visited. I will describe various reinforced random processes and their surprising properties, and outline the results achieved in this area. A special consideration will be given to the processes for which I have made a contribution, namely vertex-reinforced random walks on graphs (Ann Prob 99, 01) continuous time vertex-reinforced jump processes (PTRF 02, 04) excited random walk on trees (EJP, 03) OK Corral (JTP, 03) The talk is partly based on collaborations with Burgess Davis, Robin Pemantle, and Sir John Kingman.

|| Thursday, April 14 || || * J�nos Engl�nder, * UC Santa Barbara || || *Aspects of spatial branching in random environment * ||

When a spatial branching process is put into a random Poissonian environment, it exhibits intriguing features. We will discuss a model where "mild" obstacles affect the reproduction and will review some related similar models too.

|| Thursday, April 21 || || * Michel Bena�m, * University of Neuch�tel, Switzerland || || * Self-Interacting diffusions, dynamical systems and geometry * ||

Some models of evolution arising in economics (game theory), biology (genetic drift) and physics (polymers) are naturally described by random processes whose behavior is influenced by the full past of the process. These processes, commonly called reinforced random processes, may live on a graph (reinforced random walks) or on a manifold (self-interacting diffusion). Their non-Markovian nature makes the mathematical analysis quite challenging and most of the conjectures concerning their behaviors are still opens.

In this talk, I will mainly focus on self-interacting diffusions living on a compact manifold. I will survey recent results obtained in collaboration with O. Raimond (University Paris 11, Orsay) showing how techniques from dynamical system theory, differential topology and geometry combined with usual probabilistic tools (martingales and stochastic calculus) allow to analyze the long term behavior of these processes with a great deal of generality.

|| Thursday, April 28 || || *Dan Crisan, * Imperial College, London || || *Approximate McKean-Vlasov representations for linear stochastic partial differential equations * ||

The Wong-Zakai approximations of the (normalized) solution of a class of linear SPDEs are represented as the one-dimensional distributions of a McKean-Vlasov process. As an application, I deduce an unweighted particle representation for the solution of the stochastic filtering problem.

|| Tuesday, June 21, 2:30 PM in Van Vleck 901 || || *Paavo Salminen, * �bo Akademi University, Finland || || *Integral functionals, occupation times, and hitting times of diffusions * ||

[paavo-abstract.pdf Abstract]