Sourav Chatterjee (Berkeley)

The Komlos-Major-Tusnady embedding theorem, that gives the `best possible' coupling of a discrete random walk with a Brownian motion, is widely considered to be one of the landmark results in probability theory. The proof, however, is notoriously heavy-handed and hard to check. In this talk I will present a soft functional-analytic proof of this theorem for the case of the simple random walk. The new technique, inspired by Stein's method of normal approximation, is applicable to any setting where Stein's method works. In particular, one can hope to take it far beyond sums of independent random variables.
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Zbigniew J. Jurek (University of Wroclaw)

We will show that the reciprocal of partition functions, of a finite number of sites in one-dimensional Ising model, viewed as functions of external field are characteristic functions of selfdecomposable distributions (class L distributions). Furthermore, those distributions can be realized as stopped random walks in random environment. (cf. Reports on Math. Physics 47, 2001.)
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Jessica Zuniga (Stanford)

In this talk we will discus the quantitative analysis concerning the asymptotic behavior of time inhomogeneous finite Markov chains. To study this behavior, we develop singular value techniques in the context of time inhomogeneous chains and introduce the notion of c-stability, which can be viewed as a generalization of the case when a time inhomogeneous chain admits an invariant measure. We describe some examples where these techniques yield quantitative results for time inhomogeneous chains.
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Persi Diaconis (Stanford)

One dependent processes arise in lots of natural problems, In the carries when adding a list of numbers, descents in permutations under a variety of distributions, two-block processes in Ergodic theory and elsewhere. In joint work with Jason Fulman and Alexei Borodin we show that any one block binary processes is determinantal and that natural processes have natural kernels. I will also explain the amazing connection with quadratic algebras.
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Allan Sly (Berkeley)

The problem of reconstruction on trees has been studied in biology, information theory and statistical physics. Applications of understanding reconstruction thresholds include determining when phylogenetic reconstruction is possible and understanding the replica symmetry breaking phase transitions of glassy systems. However, exact thresholds for reconstruction on trees have only been proven for a very small number of channels. Using non-rigorous statistical physics methods Mezard and Montanari made a series of predictions for the q-state symmetric channels. I will present some recent work which establishes many of their predictions including the first exact threshold for a non-binary channel.
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Tobias Friedrich (Berkeley)

Random walks can be simulated with a simple deterministic process called rotor-router model. A seminal result of Josh Cooper and Joel Spencer showed that this deterministic model behaves pretty much the same as the random walk on Z^d. I will begin by surveying this and some other results showing a surprising similarity of the deterministic random walk to the classical random walk. Based on this model, in the main part of the talk I will propose and analyze a quasirandom analogue to the random broadcasting model for disseminating information in networks. It achieves similar or better broadcasting times with a greatly reduced use of random bits.
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Richard Liang (Berkeley)

The simplest models of population genetics, useful as they are in analyzing data, often have obvious shortcomings. Such models might ignore the effects of natural selection, mutation, or, as we will be concerned with in this talk, geography and migration. We will briefly look at the Wright-Fisher model of evolution of a single population; then, we will look at a so-called stepping stone model, where instead of a single population living all in one place, we model several populations living on discrete islands, with migration between the islands. It is often useful to consider these models' associated dual processes, which correspond to tracing the lineages of a current-day sample backwards through history. We will discuss these dual processes as well.

We will then discuss two models of evolution with *continuous* geography. Unlike the previous models, which describe directly the dynamics of a population evolving as time moves forward, the continuous geography models are instead defined in terms of prescribed dual processes. We will also discuss some properties of these models, such as continuity. This talk will be light on proofs.
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Kannan Soundararajan (Stanford)

A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete ``arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.
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Wenbo Li (University of Delaware)

Consider a family of probability measures $\{\nu_y : y \in \partial D\}$ on a bounded open domain $D\subset R^d$ with smooth boundary. For any starting point $x \in D$, we run a a standard $d$-dimensional Brownian motion $B(t) \in R^d $ until it first exits $D$ at time $\tau$, at which time it jumps to a point in the domain $D$ according to the measure $\nu_{B(\tau)}$ at the exit time, and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary. The resulting diffusion process is called Brownian motion with jump boundary (BMJ). The spectral gap of non-self-adjoint generator of BMJ, which describes the exponential rate of convergence to the invariant measure, is studied. In particular, we prove the so-called $2/3$-conjecture on the largest spectral gap (fastest rate of convergence) among all possible jump measures in one-dimensional setting. This is a joint work with Yuk Leung.
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Matthias Winkel (Oxford)

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour.

This is joint work with Bo Chen and Daniel Ford.
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Lenya Ryzhik (Stanford)

Weakly random media affect propagation of waves and particles over long distances in a non-trivial way. When medium correlations decay rapidly the long time behavior is reasonably well understood: both waves and particles behave, loosely speaking, diffusively, and most non-trivial phenomena happen on the same time scale. Slow decay of media correlations seems to produce an interesting effect when single particle behavior becomes non-trivial on a much shorter time scale than the multi-particle behavior. Similarly, the wave phase becomes random long before wave energy randomizes. Moreover, while some quantities acquire fractional Brownian limits because of the slow correlations decay, others keep their diffusive limits. I will describe some rigorous as well as formal results in these directions. This is a joint work in progress with T. Komorowski.
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Michel Ledoux (Université de Toulouse)

Joint laws of the eigenvalues of unitary invariant ensembles of random matrices, as well as largest particle distributions of some random growth models, admit a determinantal structure which may be analyzed by kernels of orthogonal polynomials. For classical orthogonal polynomials, simple Markov operator tools allow for an efficient analysis of the spectral measures, emphasizing in particular the universal role of the arcsine distribution. They also lead to moment recursion equations such as the Harer-Zagier formula in case of the Gaussian Unitary Ensemble describing map enumeration problems. In addition, sharp bounds on the largest eigenvalue or largest particle of the random matrix or random growth models at the rate of the limiting Tracy-Widom distribution may be deduced from these tools.
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Oana Mocioalca (Kent State)

Unlike Brownian motion, fractional Brownian motion exhibits long-range dependence. It has been argued, that phenomena like financial asset prices show long range dependence, and thus fBm has been proposed as a better model than Bm for describing stock prices. Using the tools of Malliavin calculus we describe a method for finding the distribution of the exit times in a trading the line strategy, a strategy where the trader will close his position when the stock price would reach a certain level, This is done under the assumption that the stock prices are driven by fractional Brownian motion or other irregular Gaussian processes.
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Souvik Ghosh (Columbia University)

We make an attempt at understanding the effect of long range dependence on the large deviation principle for the partial sums of an infinitely divisible process. It has been observed in certain short memory processes that the large deviation principle is very similar to that of an i.i.d sequence. Whereas, if the process is long range dependent the large deviations change dramatically. We want to see if such a phenomenon holds for infinitely divisible processes. We consider a stationary, mean zero infinitely divisible process without a Gaussian component but with exponentially light tails. The process is characterized by its shift invariant Lévy measure. With the aim of modeling long range dependence for such processes, we consider the situation where the Lévy measure is the law of the paths of an irreducible null recurrent Markov Chain with the marginals being the invariant measure of the chain. We study how the structure of this Markov chain affects the large deviation principle for the partial sums of the process.
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