Abstracts are added as they become available.
See the
Schedule page for the schedule.
Layering in the SOS model without external fields
Kenneth S. Alexander (University of Southern California)
We study the solid-on-solid interface model above a
horizontal wall in three dimensional space,
with an attractive interaction when the
interface is in contact with the wall, at low temperatures.
There is no bulk external field.
The system presents a sequence of layering transitions,
whose levels increase with the temperature,
before reaching the wetting transition.
Particle models with interaction through the center of mass
János Engländer (University of Colorado-Boulder)
Recently a number of particle models have been studied where individuals move in space and also interact via the center of the system (given by the center of mass). I will review some of my results as well as those of Gill, Balazs and Racz.
Tightness of maxima of generalized branching random walks
Ming Fang (Univeristy of Minnesota)
We study generalized branching random walks, which allow time dependence and local dependence between siblings. Under appropriate tail assumptions, we prove the tightness of $F_n(\cdot-Med(F_n))$, where $F_n(\cdot)$ is the maxima distribution at time $n$ and $Med(F_n)$ is the median of $F_n(\cdot)$. The main component in the argument is a proof of exponential decay of the right tail $1-F_n(\cdot-Med(F_n))$ using a Lyapunov function.
Large deviations in infinite dimensional stochastic analysis
Arnab Ganguly (ETH Zurich)
We consider a sequence of stochastic integrals $\{X_{n-}\cdot Y_n\}$, where $\{Y_n\}$ is a sequence of infinite dimensional semimartingales indexed by a separable Banach-space $H$ and time. The $X_n$ are $H$-valued cadlag processes. Assuming that $\{(X_n,Y_n)\}$ satisfies a large deviation principle, a uniform exponential tightness condition is described under which a large deviation principle holds for the stochastic integral $\{X_{n-}\cdot Y_n\}$. A simplified expression of the rate function for the sequence of stochastic integrals $\{X_{n-}\cdot Y_n\}$ has been given in terms of the rate function for $\{(X_n,Y_n)\}$. A similar result for stochastic differential equations has also been proved. The above results provide a new approach to the study of large deviations of Markov processes. Examples are given to illustrate the usefulness of this approach.
Large deviations for the partition functions of certain directed polymer models
Nicos Georgiou (Univeristy of Wisconsin-Madison)
We present results about large deviation rate functions for the partition function of a directed polymer in random environment. Existence and basic limiting properties are established for the general model. For the case of a 1+1 dimensional directed polymer with negative log-Gamma weights, a Burke-type property allows for explicit computations. The talk will be focused on the main ideas involved for the explicit computations.
Invariance principle for random walks in balanced random environment
Xiaoqin Guo (Univeristy of Minnesota)
Lawler (1982) proved a quenched invariance principle for
$\mathbb{Z}^d (d\ge2)$
random walks in balanced, uniformly elliptic random
environment. In this talk, we will show that the quenched invariance priciple still holds in i.i.d. balanced environment under
mere ellipticity.
This is a joint work with my advisor Ofer Zeitouni.
Random walk among random conductances
Christopher Hoffman (Univeristy of Washington)
Simple random walk on $\mathbb{Z}^d$ is one of the most studied objects in
probability. In this talk we will discuss which properties of random
walk on $\mathbb{Z}^d$ are shared by more general random walks in random
environments. In particular we consider random walk on percolation
clusters. We discuss the ways in which random walk on percolation
clusters exhibits similar behaviors as random walk on $\mathbb{Z}^d$. We will
also consider the more general class of random walk among a field of
random conductances. While this more general class shares some of the
properties of random walk on $\mathbb{Z}^d$, we will show that they can exhibit
many very different behaviors as well.
This is joint work with Noam Berger, Marek Biskup and Gady Kozma.
The chaotic character of the stochastic heat equation
Mathew Joseph (Univeristy of Utah)
I will describe some interesting properties of the Stochastic Heat equation,
focusing on intermittency (which is the appearance of rare and intense
peaks) and the spatial growth of the solution.
This is based on joint work with Daniel Conus and Davar Khoshnevisan.
Asymptotics for fast mean-reverting stochastic volatility models
Rohini Kumar (University of California-Santa Barbara)
We study the behavior of European call option prices and the corresponding implied volatilites near maturity. The underlying stock price
satisfies a stochastic differential equation where the volatility term (diffusion coefficient) is a function of a fast mean-reverting stochastic process. The mean-reversion time of the volatility process is of smaller order compared to the time to maturity. This separation of time scales leads to an averaging/homogenization type of problem as time to maturity approaches zero. Viscosity solution techniques are used to obtain a large deviation principle for stock price in small time. This is joint work with Jean-Pierre Fouque at University of California-Santa Barbara and Jin Feng at Kansas University.
Particle representations and limit theorems for stochastic partial
differential equations
Thomas G. Kurtz (University of Wisconsin-Madison)
Solutions of a large class of stochastic partial differential equations can
be represented in terms of the de Finetti measure of an infinite
exchangeable system of stochastic ordinary differential equations. These
representations provide a tool for proving uniqueness, obtaining convergence
results, and describing properties of solutions of the SPDEs. The basic
tools for working with the representations will be described. Applications
include the convergence of an SPDE as the spatial correlation length of the
noise vanishes, uniqueness for a class of SPDEs, and consistency of
approximation methods for the classical filtering equations.
Doeblin's ergodicity coefficient: lower-complexity approximation of occupancy distributions
Manuel Lladser (University of Colorado-Boulder)
We illustrate how a strictly positive Doeblin's coefficient leads to low- to moderate-complexity approximations of occupancy distributions of homogeneous Markov chains over finite state spaces, in the regime where exact calculations are impractical and asymptotic approximations may not be yet reliable. The key idea is to use Doeblin's coefficient to approximate a Markov chain of duration n by independent realizations of an auxiliary chain of duration $\mathcal O(\ln(n))$. To address the general case of an irreducible and aperiodic chain with a vanishing Doeblin's coefficient, we prove that Doeblin's coefficient satisfies a sub-multiplicative type inequality. A byproduct of this inequality is a new an elementary proof of Doeblin's characterization of the weak-ergodicity of non-homogeneous Markov chains. This research has been partially supported by NSF grant DMS #0805950.
Optimal uncertainty quantification
Houman Owhadi (California Institute of Technology)
Although Uncertainty Quantification (UQ) appears to be an umbrella term,
interactions with engineers show that behind this term lies a real challenge
of practical importance that is not addressed by current methods in
classical probability theory and statistics. We show how this challenge can
be formulated as an optimization problem over an infinite dimensional
(possibly non-separable) set of functions and measures of probability.
These problems correspond to extremizing probabilities of
failure, or of deviations, subject to the constraints imposed by the
scenarios compatible with the assumptions and information. Although these
optimization problems are extremely large, we show that under general
conditions, they have finite-dimensional reductions. As an application, we
develop
Optimal Concentration Inequalities
of Hoeffding and McDiarmid type. Surprisingly, these results show that
uncertainties in input parameters, which propagate to output uncertainties
in the classical sensitivity analysis paradigm, may not do so when the
transfer functions (or probability distributions) are imperfectly known.
This is a joint work with C. Scovel, T. Sullivan, M. McKerns and M. Ortiz.
A preprint is available at
http://arxiv.org/abs/1009.0679v1
Universality of Wigner matrices
José A. Ramírez (Universidad de Costa Rica)
We consider Wigner random matrices. Under some conditions on the density we prove that the two point correlation function for the eigenvalues in the bulk converges to the Dyson sine kernel. The proof is based upon
an approximate time reversal of the Dyson Brownian motion and a careful application of the method of steepest descent to the asymptotics of the transition probability of that process.
Phase transition and reconstruction for the Glauber dynamics on trees
Ricardo Restrepo (Georgia Institute of Technology)
We study the effect of boundary conditions on the relaxation time of
the Glauber dynamics on spin systems over trees. In this talk we will
describe an intimate relation between the slow down of the dynamics
and the existence of a reconstruction algorithm.
The talk in mainly based in joint work with Daniel Stefankovic, Eric
Vigoda, Juan Vera and Linji Yang.
http://arxiv.org/abs/1007.2255
Recurrence of a randomly perturbed random walk on the quarter-plane
Florian Sobieczky (University of Jena)
We give criteria for the recurrence/transience of nearest-neighbour random walks on the quarter-plane $\mathbb{Z}_+^2$ with some random perturbations. The case of equal transition probabilities has been
completely solved [1]. Some applications to queueing-networks are
presented [2].
[1] G. Fayolle, R. Iasnogorodski, V. Malyshev: `Random Walks in the Quarter-Plane', Springer 1999, Theorem 1.2.1
[2] F. Sobieczky, G. Rappitsch, G., E. Stadlober: `Tandem queues for inventory management under random perturbations', Quality and Reliability Engineering International, doi: 10.1002/qre.1161
The rate of escape on some solvable groups
Russ Thompson (Cornell University)
We exhibit random walks on certain solvable groups whose expected distance from the origin behaves like $\sqrt{n}$. We will discuss how this result relates to the underlying geometry of certain subgroups and to random walks on lamplighter groups.
Some results and conjectures for tree polymers under weak & strong disorders
Edward C. Waymire (Oregon State Univeristy)
Tree polymers are simplifications of
1+1 dimensional lattice polymers made up
of polygonal paths of a (nonrecombining)
binary tree having random path probabilities.
The path probabilities are (normalized) products of i.i.d.
positive weights. The a.s. probability laws of these
paths are of interest under weak and strong types of
disorder. Some recent results, speculation and conjectures
will be presented for this class of models under both weak
and strong disorder conditions. In particular results are included
that suggest an explicit formula for the asymptotic variance of
the "free end" of a polymer under strong disorder. This is based
on joint work with Stanley Williams and Torrey Johnson.
An integral equation driven by a FBM
Chandana Janaka Wijeratne (Univeristy of Wyoming)
An integral equation representing a vortex filament is studied. This SDE is driven by a FBM with parameter $H>1/2$ and a global existence and uniqueness is proved for a mollified version of the equation.