Probability and Mathematical Physics Seminar

This talk is a survey of some recent fluctuation results for three models: ballistic random walk in random environment, the random average process, and the asymmetric simple exclusion process. We will see fluctuations of different magnitudes, expressed as powers of the ratio between macroscopic and microscopic space and time scales.

Consider n positive diffusions whose logarithms are Brownian\ motions whose drift vector at every time point is determined by the arrangement of indices in increasing order of values. These processes appear naturally in a variety of areas from queueing theory, statistical physics, and economic modeling. For finite n, the invariant distribution of the vector of spacings between the Brownian particles can be completely described. The interest is to describe a limiting invariant distribution when n is large. We show, as n grows to infinity, a curious phenomenon occurs for the rescaled positive diffusions divided by the sum of their coordinate values. Under very weak conditions, one of three things can happen to the scaled values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to a Poisson-Dirichlet point process. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. The other alternative is to start with a countable collection of diffusions. We consider one such model and discuss the similarities and differences with the previous limit. This countable model is related to the Harris model of elastic collision and the discrete Ruzmaikina-Aizenmann model for competing particles. This is based on separate joint works with Sourav Chatterjee and Jim Pitman.

We study a long standing open problem on the mixing time of Kac's random walk on SO(n, R) by random rotations.  We obtain an upper bound mix = O(n^{2.5} log n) for the weak convergence which is close to the trivial lower bound O(n^2). This improves the upper bound O(n^4 log n) by Diaconis and Saloff-Coste. The proof is a variation on the coupling technique we develop to bound the mixing time for compact Markov chains.

An allocation rule for the standard Poisson point process in R^d is a translation-invariant way of allocating to the Poisson points mutually disjoint cells of volume 1 that cover almost all R^d. I will describe a new construction in dimensions 3 and higher of an allocation rule based on Newtonian gravitation: each Poisson point is thought of as a star of unit mass, and the cell allocated to a star is its basin of attraction with respect to the flow induced by the total gravitational force exerted by all the stars. This allocation rule is efficient, in the sense that the distance a typical point has to move is a random variable with exponentially decreasing tails.
The talk is based on joint work with Sourav Chatterjee, Ron Peled and Yuval Peres.

3:45 P.M., WWH 1302
Two-dimensional Polymers (self-avoiding walks) and Their Continuum Limit 
Greg Lawler, University of Chicago]

Stein's method of exchangeable pairs is a powerful tool for proving rates of convergence in probabilistic limit theorems.  It is usually implemented in conjunction with some inherent symmetries of the random variable of interest. I will discuss a new version of the method (based on ideas of Charles Stein) for situations in which the underlying symmetries used are of a continuous nature. The method has applications is random matrix theory in proving the asymptotic normality of linear functions on the classical compact matrix groups, and in Riemannian geometry, in studying the value distributions of eigenfunctions of the Laplacian.  I will also discuss a multivariate version of the method.

The mathematical field of queueing theory was introduced in the first half of the 20th century to model voice communication networks. In the second half of the 20th century, queueing theory was also applied to data communication systems and contributed to the design of the first prototype for the Internet. Both types of voice and data queueing models made significant use of the steady state theory for continuous time Markov chains.
This talk discusses the new types of mathematical tools needed to create a dynamical queueing theory. This involves methods such as Poisson random measures as well as perturbation methods that are applied to both the differential equations for the transition probabilities and the sample path behavior of time inhomogeneous Markov chain queueing models. These tools help us to capture more of the dynamic time-varying behavior of these systems that would otherwise be washed out by steady state analysis.
Finally, we can establish fundamental limit theorems that approximate many of these random processes by dynamical systems. From these results, we can apply the dynamic optimization techniques of classical mechanics to the efficient design of these queueing models.

We will present the DLA model, and comment on few versions.

Princeton University, Computer Science Building auditorium]

3:30 P.M., WWH 109
Traces, determinants and probability theory
Jean-Michel Bismut, Université Paris-Sud and Courant Institute]

Universality limits arise in random matrix theory, and their proof involves asymptotics for orthogonal polynomials. The Riemann-Hilbert techniques have yielded powerful forms of these, with complete error estimates. We show that a localization and smoothing technique permits proof of (first order) universality limits both in the bulk, and at the edge of the spectrum, under minimal assumptions on the weight. For example, in the case of a fixed weight on [-1,1], all one needs is continuity at the points where universality is desired.

The totally asymmetric simple exclusion process (TASEP) is one of the simplest models of interacting particle systems on the one-dimensional lattice. It is equivalent to a random growth model from the Kardar-Parisi-Zhang universality class. We focus on fluctuations of the particle positions for a nonequilibrium TASEP that starts from certain deterministic initial conditions. We (rigorously) derive the scaling exponents 1/3 and 2/3, and identify the limit laws as those of Gaussian Orthogonal and Unitary ensembles of the random matrix theory. The emphasis will be made on a new approach that circumvents the Robinson-Schensted-Knuth algorithm -- a nontrivial combinatorial construction that has so far played a central role in the analysis of similar models.

We report on some recent work with Itai Benjamini where percolative properties of the vacant set left by simple random walk on a discrete torus of large side-length by times of order the number of sites in the torus are investigated. We also discuss some related results concerning the disconnection time of discrete cylinders obtained in an other recent work in collaboration with Amir Dembo.

3:45 P.M., WWH 1302
Random Motions in Random Media
Alain-Sol Sznitman, ETH Zurich]

The sine and Airy point processes arising from random matrix eigenvalues play a fundamental role in probability theory, partly due to their connection to Riemann zeta zeros and random permutations. I will describe recent work on the Stochastic Airy and Stochastic sine differential equations, which are shown to describe these point processes and can be thought of as scaling limits of random matrices. These equations can be thought of as random Schroedinger operators.

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. We let $n$ go to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the $k$-core is empty, and other conditions that imply that with high probability the $k$-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald \cite{psw96} on the existence and size of a $k$-core in $G(n,p)$ and $G(n,m)$, see also Molloy~\cite{Molloy05} and Cooper~\cite{c04}.
Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. We develop that method further and show that the fluctuations around the deterministic limit converge to a Gaussian law above and near the threshold, and to a non-normal law at the threshold. Further, we  determine precisely the location of the phase transition window for the emergence of a giant $k$-core. Hence we deduce corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$.
This is joint work with Svante Janson.

We review recent results on scaling limits for trap models in a few graphs. In particular, we introduce K-processes, which are related to the trap model in the complete graph, as well as in the hypercube, in times of the order of the size of the deepest traps, and discuss their properties, including aging behavior.

I will first introduce the classical "Becker-Doring" equation for the distribution of nuclei and describe its time-dependent asymptotic solution for a high nucleation barrier. Then, I will discuss non-classical nucleation in a two-dimensional Ising model driven by Glauber/Metropolis dynamics. Here, accurate values of the nucleation rate can be derived at low-temperatures from first-principle considerations and used to assess the phenomenological Becker-Doring picture.

The Ruelle Probability Cascades (RPC) are stochastic objects consisting of a random atomic measure on the half-line together with a superimposed distance between the atoms. Objects of this kind are a natural setting to study the evolution of a configuration of points under correlated shift, the shift being a function of the superimposed distance between points. It turns out that the RPC's are stable under such evolution. In this talk, we will define precisely the RPC's and derive their stability property. Secondly, we will see how they appear in the study of mean-field spin glass systems and discuss some related open questions.

We give bounds on the diffusivity of finite-range asymmetric exclusion processes on Z with non-zero drift.  We use the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the recent works of Ferrari and Spohn, and Balazs and Seppalainen provide sharp bounds.

We prove hydrodynamic limit for the simple exclusion process with degenerated bond disorder. The hydrodynamic equation corresponds to the generator of a generalized diffusion with diffusion coefficient given by a stable subordinator.

I recall the definition of restricted solid-on-solid models in statistical mechanics, which are generalizations of the two-dimensional Ising model where the degrees of freedom take values on the nodes of a Coxeter diagram of type A, D or E, and which have local weights. I show that the law of suitably defined cluster boundaries in these models is identical with that of the corresponding curves in the (non-local) O(n) or Q-state Potts models, and they are therefore conjectured to be described in the scaling limit by SLE(kappa) with kappa rational and >2.

Families of ergodic finite Markov chains indexed by the size/complexity of the system often present a cutoff: convergence occurs abruptly in a time window much shorter than the mixing time itself. For such families, the notion of ``time to equilibrium'' or ``mixing time'' is meaningful, e.g., random transpositions of n cards take (1/2)n log n to mix up the cards. Deciding when a given family of chains presents a cutoff and why has proved to be an elusive problem. After introducing the notion of cutoff, I will discuss a number of examples and results related to this question.

I will describe some results concerning Dyson's circular (random matrix) ensembles (i.e., the Coulomb gas on the unit circle) at arbitrary temperature.

Nonlinear Schrodinger equations are a generic model for the propagation of nonlinear dispersive waves, eg in fibers, crystals, Bose-Einstein condensates... It is often physically relevant to consider random perturbations of these equations. We present large deviations for various kinds of small noises. We then review some applications to the blow-up times and physical quantities whose fluctuation due to noise impair transmission by solitons in fibers. We finally present the problem of the exit of a domain of attraction for weakly damped equations and the associated variational problem.

On a compact Riemannian manifold, minus t times the logarithm of the heat kernel converges uniformly to the energy function as t goes to zero. This limit commutes with spatial derivatives away from the cut locus, but one expects more complicated behavior at the cut locus. We will give formulas for the small time asymptotics of the gradient and the Hessian of the logarithm of the heat kernel which are valid everywhere and which admit an appealing probabilistic interpretation. We will also show how these formulas can be used to study both the pointwise and the distributional limits of derivatives of the logarithm of the heat kernel.

We shall review the recent progress on the derivation of the Gross-Pitaevskii Equation for the Bose-Einstein Condensate. Besides introduce the concept of Bose-Einstein condensate, the main focus of the talk is to review the tools available in N-body quantum dynamics. This includes BBGKY hierarchy, method of moments of energy and Feynman diagrams. We shall also briefly review how to turn estimates on Feynman diagrams into a tool for establishing well-poseness of Schrodinger equations in infinite dimensions.

in 1990, Ball and Carr proved some roughly sharp conditions for mass conservation of a spatially homogeneous system of equations modelling binary coagulation. In joint work with Fraydoun Rezakhanlou, we present similar conditions for a system with spatial dependence and diffusive motion. The most important element in the derivation are L^{\infty} bounds on the solutions of the PDE. I will discuss a random method for deriving these bounds, which involves monitoring a tracer particle in a field specified by a solution of the PDE.