# Past Probability Seminars Fall 2012

Thursday, September 13,
Sebastien Roch, UW-Madison
Markov models on trees: Variants of
the reconstruction problem

Abstract: I will consider the so-called ``reconstruction problem*: how accurately can one guess the state at the root of a Markov chain on a finite tree, given the states at the leaves? I will introduce variants of this problem that arise naturally in connection with applications in molecular evolution, and discuss recent results and open problems. Based on joint works with Andoni, Daskalakis, Hassidim, Mossel and Sly.*

Thursday, September 20, Jun Yin, UW-Madison Some new results on random matrices Abstract: In this talk, we will introduce some new results on random matrices, especially the necessary and sufficient conditions for universality at the edge and a new result on the circular law.

Friday, October 5, 2:25, VV B211
Nicos Georgiou, University of Utah
Busemann functions and variational formula for last passage
percolation.

Abstract: Directed last passage percolation on the two dimensional lattice is exactly solvable when the weight distribution is i.i.d. exponential or geometric. The reason for that is the Burke property associated to a model with "boundaries".

We investigate the solvable model further in order to generalize the idea of boundaries into the general setting, and we compute a variational formula for passage times for more general weights. The variatonal formula is given in terms of Busemann functions and all restrictive assumptions on the environment are to guarantee their existence.

Joint work with T. Seppalainen, F. Rassoul-Agha and A. Yilmaz.

Thursday, October 11,
NO SEMINAR!
Midwest Probability Colloquium

Thursday, October 18 Jason Swanson, University of Central Florida Correlations within the signed cubic variation of fractional Brownian motion

Abstract: The signed cubic variation of the fractional Brownian motion, $B$, with Hurst parameter $H=1/6$, is a concept built upon the fact that the sequence, $\{W_n\}$, of sums of cubes of increments of $B$ converges in law to an independent Brownian motion as the size of the increments tends to zero. In joint work with Chris Burdzy and David Nualart, we study the convergence in law of two subsequences of $\{W_n\}$. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.

Thursday, October 25
Mihai Stoiciu, Williams College
Random Matrices with Poisson Eigenvalue Statistics

Abstract: Several classes of random self-adjoint and random unitary matrices exhibit Poisson microscopic eigenvalue statistics. We will outline the general strategy for proving these results and discuss other models where the Poisson statistics is conjectured. We will also explain how changes in the distribution of the matrix coefficients produce changes in the microscopic eigenvalue distribution and give a transition from Poisson to the picket fence distribution.

Thursday, November 8 Michael Kozdron University of Regina The Green's function for the radial Schramm-Loewner evolution

Abstract: The Schramm-Loewner evolution (SLE), a one-parameter family of random two-dimensional growth processes introduced in 1999 by the late Oded Schramm, has proved to be very useful for studying the scaling limits of discrete models from statistical mechanics. One tool for analyzing SLE itself is the Green's function. An exact formula for the Green's function for chordal SLE was used by Rohde and Schramm (2005) and Beffara (2008) for determining the Hausdorff dimension of the SLE trace. In the present talk, we will discuss the Green's function for radial SLE. Unlike the chordal case, an exact formula is known only when the SLE parameter value is 4. For other values, a formula is available in terms of an expectation with respect to SLE conditioned to go through a point. This talk is based on joint work with Tom Alberts and Greg Lawler.

Thursday, November 15, 2:25 pm Gregorio Moreno Flores, UW - Madison Directed polymers and the stochastic heat equation

Abstract: We show how some properties of the solutions of the Stochastic Heat Equation (SHE) can be derived from directed polymers in random environment. In particular, we show:

A new proof of the positivity of the solutions of the SHE Improved bounds on the negative moments of the SHE Results on the fluctuations of the log of the SHE in equilibrium, namely, the Cole-Hopf solution of the KPZ equation (if time allows).

Tuesday, November 27, 2:30 pm, VV 901 Michael Damron, Princeton Busemann functions and infinite geodesics in first-passage percolation

Abstract: In first-passage percolation we study the chemical distance in the weighted graph Z^d, where the edge weights are given by a translation-ergodic (typically i.i.d.) distribution. A main open question is to describe the behavior of very long or infinite geodesics. In particular, one would like to know if there are infinite geodesics with asymptotic directions, how many are there, and if infinite geodesics in the same direction coalesce. Some of these questions were addressed in the late 90's by Newman and collaborators under strong assumptions on the limiting shape and weight distribution. I will discuss work with Jack Hanson (Ph. D. student at Princeton) where we develop a framework for working with distributional limits of Busemann functions and use them to prove a form of coalescence of geodesics constructed in any deterministic direction. We also prove existence of infinite geodesics which are asymptotically directed in sectors. Last, we introduce a purely directional condition which replaces Newman's global curvature condition and whose assumption implies the existence of directional geodesics.

Thursday, December 6
Scott McKinley University of Florida
Sensing and Decision-Making in Random Search

Abstract: Many organisms locate resources in environments in which sensory signals are rare, noisy, and lack directional information. Recent studies of search in such environments model search behavior using random walks (e.g., Levy walks) that match empirical move- ment distributions. We extend this modeling approach to include searcher responses to noisy sensory data. The results of numerical simulation show that including even a simple response to noisy sensory data can dominate other features of random search, resulting in lower mean search times and decreased risk of long intervals between target encounters. In particular, we show that a lack of signal is not a lack of information. Searchers that receive no signal can quickly abandon target-poor regions. On the other hand, receiving a strong signal leads a searcher to concentrate search effort near targets. These responses cause simulated searchers to exhibit an emergent area-restricted search behavior similar to that observed of many organisms in nature.

Thursday, December 13
Karl Liechty, University of Michigan
Extremal statistics of the Airy2 process minus a parabola

Abstract: For a directed polymer in a random medium in the point-to-line geometry, both the fluctuations of the energy and of the position of the polymer can be described in terms of the Airy2 process. The energy fluctuations are described by the maximum of Airy2 process minus a parabola, and the fluctuations in the location of the endpoint are described by the location of this maximum. It is known that the maximum of Airy2 process minus a parabola is described by the Tracy-Widom GOE distribution, but somewhat less is known about the location of the maximum. I will discuss recent work in this area, focusing on the approach to the problem which is based on analysis of orthogonal polynomials.