# Past Probability Seminars Spring 2004

## UW Math Probability Seminar Spring 2004

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 29 || || * Marek Biskup,* UCLA || || * Graph distance in long-range percolation models * ||

In 1967, using an ingenious sociological experiment, S. Milgram studied the length of acquaintance chains between geometrically distant individuals. The results led him to the famous conclusion that average two Americans are about six acquaintances (or six handshakes) away from each other. We will model the situation in terms of long-range percolation on Z^d, where the nearest neighbor bonds represent the acquaintances due to geometric proximity-people living in the house next door-while long bonds are acquaintances established by other means-e.g., friends form college. The question is: What is the minimal number of bonds one needs to traverse to get from site x to site y.

Thus, in addition to the usual connections between nearest neighbors on Z^d, any two sites x,y in Z^d at Euclidean distance |x-y| will be connected by an occupied bond independently with probability proportional to |x-y|^{-s}, where s>0 is a parameter. Using D(x,y) to denote the length of the shortest occupied path between x and y, the main question boils down to the asymptotic scaling of D(x,y) as |x-y|\to\infty. I will discuss a variety of possible behaviors and mention known results and open problems. Then I will sketch the proof of the fact that, when s\in(d,2d), the distance D(x,y) scales like (\log|x-y|)^\Delta, where \Delta^{-1} is the binary logarithm of 2d/s.

|| Thursday, February 5 || || * Lea Popovic, * IMA, University of Minnesota || || * Asymptotic genealogical process - a point-process representation * ||

We consider a continuous-time binary branching process conditioned to have population size n at some time t, and with a chance p for recording each extinct individual in the process. Within the family tree of this process, we are interested in the smallest subtree containing the genealogy of the extant individuals together with the genealogy of the recorded extinct individuals. We introduce a novel representation of such subtrees in terms of a point-process, and give asymptotic results for the distribution of this point-process as the number of extant individuals increases. We motivate the study within the scope of a coherent analysis for an a-priori model for macroevolution.

|| Thursday, February 12 || || * Timo Sepp�l�inen, * UW-Madison || || * Phase transition behavior of disordered asymmetric exclusion * ||

In a disordered asymmetric exclusion process we give each particle its own jump rate, independently drawn from a distribution F. When F has a thin enough tail at its left end, the system has a critical density below which there are no equilibrium distributions. In this talk I will review what is known about this model and present some recent results obtained jointly with Ilie Grigorescu and Min Kang.

|| Thursday, February 19 || || *Chris Henley, * Physics, Cornell University || || * Open Problems in Random Tiling Quasicrystals * ||

A "random tiling" ensemble is defined, roughly, as a set of tilings of 2- or 3-space such that the number of tilings N_t grows exponentially with the volume V covered. A "quasicrystal" random tiling ensemble means that the set of tiles has a symmetry (e.g. 10-fold rotations) which is incompatible with a (non-dense) lattice. The random tiling scenario is one of the two competing paradigms of the real "quasicrystal" alloys, and offers predictions about diffraction (i.e. Fourier spectrum of the vertices) which are supported by experiments.

I will briefly review my nonrigorous statistical-mechanics studies, based on the "random tiling hypotheses" which relate the entropy density, log (N_t)/V) in limit of infinite V, to the "phason strain", which quantifies the deviation of the tile frequencies from symmetry. I will also describe the most interesting opportunities for rigorous mathematical approaches to quasicrystal random tilings. This includes a survey of

(i) the "exact solutions" for the tiling of squares and equilateral triangles (and two related tilings), by Widom, Kalugin, de Gier, and Nienhuis, using the Bethe Ansatz method
(ii) close packings of hard disk mixtures which form random tilings
(iii) tiling sub-ensembles defined by maximizing the frequency of selected patterns
(iv) tilings of rhombi with p-fold symmetry, in the limit of large p, as studied by Mosseri, Destainville, and Widom.
(v) problems in the accessibility of one tiling from another in Monte Carlo simulations


|| Thursday, February 26 || || * Antar Bandyopadhyay, * IMA, University of Minnesota || || * Recursive Distributional Equations and Associated Recursive Tree Processes * ||

In a variety of applied probability settings, from the study of Galton-Watson branching processes to mean-field statistical physics models, a central theme is to solve some fixed-point-equation on an appropriate space of probabilities, we call it a Recursive Distributional Equation (RDE). Exploiting the natural recursive structure one can associate to such an equation a much richer probability model which we call a Recursive Tree Process (RTP). In some sense if a RDE has a solution then the corresponding RTP is an almost sure representation of it. In this talk we will outline some basic general theory related to such tree indexed processes with main focus on the question on endogeny : the RTP being measurable with respect to the associated innovation process (the date available from the basic RDE). We will state necessary and sufficient conditions for endogeny which can be used in a variety of context. As an application we will specialize to the question of measurability of the frozen percolation process on infinite binary tree and will prove that the process as constructed by Aldous (2000) is measurable with respect to the edge weights. Throughout the talk this example will be used to illustrates the concepts. (This is a joint work with Professor David J. Aldous)

|| Thursday, March 4 || || * Rick Kenyon, * Princeton University || || * Limit shapes of crystalline surfaces * ||

(This is joint work with Andrei Okounkov.) We study a simple model of crystalline surfaces in R^3. These come from limits of discrete interfaces in the dimer model (domino tiling model). These discrete interfaces can be viewed as a higher-dimensional generalization of the simple random walk, where the domain is (part of) Z^2 instead of Z. We are interested in the behavior of these interfaces in the scaling limit (limit when the mesh tends to zero): the limit surfaces minimize a certain surface tension functional which arises from purely entropic considerations. Remarkably, the limit surfaces, which are solutions of a nonlinear PDE, can be parametrized by analytic functions and may contain facets in certain rational directions.

|| Thursday, March 11 || || * * || || * * ||

|| Thursday, March 18 || || SPRING BREAK ||

|| Thursday, March 25 || || * Noam Berger, * Caltech || || * Biased random walk on percolation clusters * ||

We consider biased random walk on supercritical percolation clusters in Z^2. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. This is joint work with Nina Gantert and Yuval Peres.

|| Thursday, April 1 || || * James Kuelbs, * UW Madison || || * Weighted occupation measures and escape rates for certain processes * ||

[kuelbs-abs.pdf Abstract]

|| Thursday, April 8 || || * Firas Rassoul-Agha, * Ohio State University || || * On the 0-1 Law and the Law of Large Numbers for Random Walks in Mixing Random Environments * ||

We prove a weak version of the law of large numbers for multi-dimensional finite range random walks in certain mixing elliptic random environments. We also prove that for such walks, the zero-one law implies a strong law of large numbers.

|| Thursday, April 15 || || * Craig Tracy, * University of California-Davis || || * Differential equations for Dyson diffusion * ||

|| Thursday, April 22 || || * C�cile An�, * University of Paris XI and UC Davis || || * Logarithmic Sobolev inequalities for Markov processes on graphs * ||

In this talk, I investigate some properties of the distribution of the path of Markov processes. Results obtained for Markov processes on graphs are compared to what is known for Brownian motion and diffusions. I will discuss functional inequalities, namely Poincar� and logarithmic Sobolev inequalities. They bound the variance or entropy of a function by its energy, a quantity involving the derivative of that function. The main tool for deriving these inequalities is a Clark-Ocone formula. Different discrete derivatives on path space will be discussed.

|| Thursday, April 29 || || * Michael Cranston,* University of Rochester || || * Some recent results on the parabolic Anderson model * ||

The parabolic Anderson model refers to parabolic pdes of the form k\Delta u+ V u = u_t, where \Delta is the Laplacian on Z^d or R^d and V is a random potential. This equation models many phenomena such as entrapment of electrons in crystals with impurities (Anderson's original motivation), population dynamics with random births and deaths and magneto-hydrodynamics. It is also related by simple transformations to Burger's equations and to the KPZ equation for random surface growth. In this talk we shall discuss the asymptotic behavior of solutions of the equation for various choice of potential V . We shall show how they depend on the parameter k as k tends to 0. Traditionally, the equation has been considered over Z^d, but we will also discuss some results where the underlying state space is R^d.