See the Schedule page for the schedule.

A spectral analysis of the sequence of firing phases in stochastic Integrate-and-Fire oscillators
Peter Baxendale (University of Southern California)

The integrate and fire oscillator is a widely used model for the evolution of the membrane potential in a nerve cell. One important problem is to determine the effect of periodic modulation of the input current or the firing threshold function on the sequence of firing times. For a noise-free system the sequence of firing phases (modulo the period of the modulation) is a deterministic dynamical system on the circle, and its bifurcation scenario has been studied by many authors. As soon as white noise is added to the membrane potential, the sequence of firing phases becomes a uniformly ergodic Markov chain on the circle and the bifurcation behavior is "smoothed out". However numerical computations with small noise intensity suggest that some of the deterministic behavior shows up in the eigenvalues of the Markov transition operator. This talk will describe recent theoretical results, obtained jointly with John Mayberry (Cornell University), on small noise asymptotics of the spectrum of this Markov transition operator.

Inverse problems for stochastic PDEs
Igor Cialenco (Illinois Institute of Technology)

We will discuss a parameter estimation problem for a some stochastic evolution equation driven by additive or multiplicative space-time white noise. Some asymptotic properties of the maximum likelihood estimator will be investigated, and necessary and sufficient conditions for consistency and asymptotic normality will be presented.

The asymmetric simple exclusion process
Ivan Corwin (Courant Institute, New York University)

The Asymmetric Simple Exclusion Process is a Markov Process used to model numerous physical and mathematical systems. In this short talk we introduce the process and explain some recent results about the current fluctuations of the process.

A solvable homopolymer model
Michael Cranston (University of California, Irvie)


An interacting branching model
János Engländer (University of California, Santa Barbara)

I will present a model where a discrete particle system has both branching and interaction and a result on its large time behavior. The method involves the study of the motion of the center of mass which is extended for superprocesses as well.

Probability fringe convergence and the eigenvalues of large random trees
Steven N. Evans (University of California, Berkeley)

We describe recent research on the eigenvalues of quite general models of (large) random rooted trees, including various of the preferential attachment schemes that have recently attracted interest in computer science as models of real-world networks, as well as random recursive trees, Yule trees, and uniform random trees.

Many such ensembles possess the property of probability fringe convergence. Very roughly speaking, this means that the neighborhood of a uniformly chosen vertex converges in distribution as the size of the tree increases, and the neighborhoods of two uniformly chosen points are asymptotically independent. The usual tool for proving probability fringe convergence is to embed the random trees of interest into a suitable continuous time-branching process.

We use fairly simple ideas from linear algebra (primarily the interlacing inequalities) to show that if a sequence of random trees converges in the probability fringe sense, then the empirical distributions of the eigenvalues of the corresponding adjacency matrices converge in distribution to a deterministic limit. Moreover, the masses assigned by the empirical distributions to individual points also converge in distribution to constants. We conclude for ensembles such as the linear preferential attachment models and the random recursive trees that the limiting spectral distribution has a set of atoms that is dense in the real line. However, we are unable to tell whether the limiting spectral distributions are purely atomic. One technically interesting feature of this work is that the most commonly used tool in random matrix theory, the method of moments, cannot be applied to some natural ensembles, because the expected values of higher moments of the empirical spectral measure diverge.

We were able to get precise asymptotics on the mass assigned to zero by the empirical spectral measures due to the connection between the number of zero eigenvalues of the adjacency matrix of a tree and the cardinality of a maximal matching on the tree.

In particular, we use a simplified version of an algorithm due to Karp and Sipser to construct maximal matchings and understand their properties.

We also established the joint convergence in distribution of the suitably normalized k largest eigenvalues of the linear preferential attachment ensemble for any k using a previously known connection between the largest eigenvalues of this model and the largest out-degrees.

This is joint work with Shankar Bhamidi (U. of British Columbia) and Arnab Sen (U.C. Berkeley).


Error analysis of simulations for a jump type Markov process
Arnab Ganguly (University of Wisconsin, Madison)

Here we try to analyze the error of a special type of Jump Markov process. These type of Markov Processes arise in modeling system of chemical reactions where the state of the system, that is the vector of number of molecules of the reactant species, is given by some SDE, which we call the "exact equation". There are algorithms to simulate from the exact equation but for a large system they are quite slow. So people try to bring in some approximations to the exact equation and developed some faster algorithms called "Tau Leap method", and the "The Midpoint Method". I will discuss these two methods in my talk, and the main goal is to try to analyze the error arising from these approximations and also to compare the accuracy of Tau Leap and The Midpoint Method.

Soft edge results for longest increasing paths on the planar lattice
Nicos Georgiou (University of Wisconsin, Madison)

For two-dimensional last-passage time models of weakly increasing paths several results have been found about the hard edge (close to axis). For strictly increasing paths of Bernoulli(p) marked sites, questions about the hard edge are addressed using SLLN. We are interested in the maximal cardinality of marked sites on a strictly increasing path, in a rectangle of dimensions [p-1n-xna]×n and we present (soft) edge results for such paths in this rectangle.

Stochastic model for concentration in yeast cell
Ankit Gupta (University of Wisconsin, Madison)

We study the model proposed by Altschuler, Angenent and Wu for the dynamics of particles in a yeast cell. In this model there are two types of particles, A and B, which interact with each other. We are specifically interested in the clustering of particles on the membrane. Each cluster is called a clan. For any finite population size N, clan sizes follow a Markov Chain over the space of measures. We show that under suitable scaling these clan sizes follow a measure-valued diffusion process in the infinite population limit. Moreover the ratio of A and B type particles converges to a constant for every clan. We also obtain the stationary distribution for clan sizes.

Independent particles in a dynamical random environment
Mathew Joseph (University of Wisconsin, Madison)

We study the motion of independent particles in a dynamical random environment on the integer lattice. We will show that the spatially ergodic invariant distributions for the process are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment.

In the second part of the talk, we will consider the fluctuations of the net current seen by an observer traveling at a deterministic speed in the one dimensional case. This is work currently in progress. We will describe some of the motivation behind the problem and also some previous results.

This talk is based on an ongoing project with Prof. Timo Seppäläinen.


On orthogonality in probability
Yevgeniy Kovchegov (Oregon State University)

For a class of simple stochastic processes, we will apply the spectral approach of Karlin-McGregor diagonalization. We will explore the various connections between random walks in random environments, Riemann-Hilbert problems, occupation times and convergence rates.

Current fluctuations for independent random walks in multiple dimensions
Rohini Kumar (University of Wisconsin, Madison)

I will talk about current fluctuations in a system of asymmetric random walks in multiple dimensions. The current process of interest is defined as a distribution valued process. Scaling time by n and space by √n gives current fluctuations of order nd/4 where d is the space dimension. The scaling limit of the normalized current process is a distribution valued Gaussian process with given covariance. It can also be expressed as the solution of a stochastic partial differential equation.

Multifractal analysis of the Schramm-Loewner evolution
Gregory F. Lawler (University of Chicago)

The Schramm-Loewner evolution (SLE) is a conformally invariant process invented by Oded Schramm as a candidate for the scaling limit of planar lattice models in two-dimensional statistical physics. It gives a random fractal curve. To understand the evolution of the curve at time t, one needs to study the derivative near the tip of the path of the conformal map that sends the slit domain to the upper half plane. I will discuss the behavior of this derivative using the reverse-time Loewner flow and give results that can be proved from this analysis, e.g.,
  • An alternative proof of Beffara's theorem on the Hausdorff dimension of the paths
  • A parametrization of the curve in a way that matches the fractal structure and should be the limit of the natural parametrization on discrete objects (joint work with Scott Sheffield)
  • The exact Holder continuity of the paths in the capacity parametrization and a multifractal spectrum relating to the local derivatives (joint work with Fredrik Johansson).

    • On uniform ergodicity for state dependent single class queueing networks
    Chihoon Lee (Colorado State University)

    We consider single class queueing networks with state dependent arrival and service rates. Under uniform (in state) stability condition, it is shown that the queue length process is $f$-uniformly ergodic that is, it has transition probability kernel which converges to its limit geometrically quickly in $f$-norm sense. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, functional central limit result, and Strassen-type law of the iterated logarithm result.

    Limit theorems for Parrondo's paradox
    Jiyeon Lee (Yeungnam University)

    That two losing games can be combined to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of profit-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios formulas for the mean permit an analysis of when the Parrondo effect is present. (Joint work with S. N. Ethier of University of Utah)

    Diffusion in soft matter
    Scott McKinley (Duke University)

    Stochastic models for diffusion of Brownian particles in soft matter (viscoelastic media) play a central role in polymer dynamics and rheology, microrheology, and medical science. A sufficiently robust class of stochastic processes is required to capture the range of observed anomalous diffusive behavior, in particular transient power law scaling of the mean-squared displacement (MSD) of tracked particles. We consider the Generalized Langevin Equation characterized by a Prony series approximation to the relaxation kernel, and study in particular this system in its zero mass limit. Such a study reveals a robust class of models which exhibit transient anomalous diffusion with a scaling law exactly expressible in terms of a parameter characterizing the relaxation spectrum of the GLE. (Joint work with Greg Forest, UNC-Chapel Hill and Lingxing Yao, Utah)

    Ising Euclidean fields and cluster area measures
    Charles M. Newman (Courant Institute, New York University)

    I will discuss a representation for the magnetization field of the critical two-dimensional Ising model in the scaling limit as a random field using renormalized area measures associated with SLE (Schramm-Loewner Evolution) clusters. The renormalized areas come from the scaling limit of critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. The representation is based on the interpretation of the lattice magnetization as the sum of the signed areas of clusters. If time permits, potential extensions, including to three dimensions will also be discussed. The talk will be based on joint work with F. Camia (arXiv:0812.4030; to appear in PNAS) and on work in progress with F. Camia and C. Garban.

    The spread of biparental ancestry
    Peter Ralph (University of California, Berkeley)

    Consider the full (biparental) genealogy of a population of sexually reproducing individuals. The genealogy of each single gene is well-understood (it is Kingman's coalescent), and is contained within this network, but the properties of the whole network are not. We use interacting particle systems and an identity from branching processes to demonstrate how the expected proportion of ancestral genes is distributed across the population at a distant time.

    Spatial and non spatial stochastic models for immune response
    Rinaldo Schinazi (University of Colorado, Colorado Springs)

    We consider a model in which every individual can give birth to an individual of the same type or to a mutated individual. Each type lives a mean 1 exponential time and then all the individuals of a given type are killed simultaneously. We compare a non spatial version of this model to versions on the square lattice and on the homogeneous tree.

    Fluctuation bounds for a class of zero range processes
    Timo Seppäläinen (University of Wisconsin, Madison)

    We look at the fluctuations of the particle current in stationary one-dimensional asymmetric particle systems with nonlinear flux. It is expected that the current seen by an observer traveling at the characteristic speed has fluctuations of magnitude t1/3 and limits that obey Tracy-Widom related distributions. The correct order of magnitude (in the sense of variance bounds) is known for asymmetric exclusion processes and some flavors of zero range and bricklayer processes. For exclusion processes exact distributional limits are also known. This talk discusses the case of zero range processes. We explain how the variance bound for the current follows from superdiffusive moment bounds for a second class particle. The proofs rely on coupling constructions. (Joint work with Márton Balázs and Júlia Komjáthy, Budapest.)

    Phase transition and universality for homopolymers based on stable processes
    Nicola Squartini (University of California, Irvie)

    We consider a polymer measure based on a Markov process which is attracted to a stable random variable. A Gibbs measure is defined which models an attraction to the origin for these process. A phase transition occurs as the the strength of the attraction to the origin increases. We examine various "thermodynamic" quantities and show they are all related to each other in a simple way and exhibit universality.