# On diffusions interacting through their ranks

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We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint work with Amir Dembo, Tomoyuki Ichiba, Soumik Pal and Ofer Zeitouni

# Effective Dynamics of Stochastic Partial Differential Equations

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The need to take stochastic effects into account for modeling complex systems has now become

widely recognized. Stochastic partial differential equations arise naturally as mathematical

models for multiscale systems under random influences. We consider macroscopic dynamics of

microscopic systems described by stochastic partial differential equations. The microscopic

systems are characterized by small scale heterogeneities (spatial domain with small holes or

oscillating coefficients), or fast scale boundary impact (random dynamic boundary condition),

among others.

Effective macroscopic model for such stochastic microscopic systems are derived. The effective

model s are still stochastic partial differential equations, but defined on a unified spatial domain

and the random impact is represented by extra components in the effective models. The

solutions of the microscopic models are shown to converge to those of the effective macroscopic

models in probability distribution, as the size of holes or the scale separation parameter

diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of

convergence in probability distribution, and in the sense of convergence in energy are also

proved.

# Competing first passage percolation on random regular graphs

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In this talk we will consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy roughly N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. These results are in sharp contrast with the picture in the lattice case.

This is a joint work with Yael Dekel, Elchanan Mossel and Yuval Peres.

# Levy concentration inequalities in higher dimensions.

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# "Subgaussian concentration and rates of convergence in directed polymers."

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# Fixation for distributed clustering processes.

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We study a discrete-time resource flow in Z^d, where wealthier vertices attract the resources of their less rich neighbors. For any translation-invariant probability distribution of initial resource quantities, we prove that the flow at each vertex terminates after finitely many steps. This answers (a generalized version of) a question posed by Van den Berg and Meester in 1991. The proof uses the mass-transport principle and extends to other graphs.

# Particle flow and negative dependence in the Symmetric Exclusion Process.

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Abstract: I'll talk about the recently discovered strong negative dependence properties of the symmetric exclusion process, a model of non-intersecting random walkers. The negative dependence theory gives a simple way to show central limit theorems for the bulk motion of particles. Our results are general enough to deal with non-equilibrium systems of particles with inhomogeneous hopping rates.

# "Ballisticity conditions for random walk in random environment"

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BALLISTICITY CONDITIONS FOR RANDOM WALK IN

RANDOM ENVIRONMENT

ALEJANDRO F. RAMIREZ

resumen. Consider a Random Walk in a Random Environment (RWRE)

{Xn :

n

&#8805; 0} on a uniformly elliptic i.i.d. environment in dimensions d &#8805; 2. Some

fundamental questions about this model, related to the concept of ballisticity

and which remain unsolved, will be discussed in this talk. The walk is said to

be transient in a direction l

&#8712; S

d

, if limn

&#8594;&#8734; Xn l = &#8734;, and ballistic in the

direction l if lim inf n

&#8594;&#8734; Xn l/n > 0. It is conjectured that transience in a

given direction implies ballisticity in the same direction. To tackle this question,

in 2002, Sznitman introduced for each &#947;

&#8712; (0, 1) and direction l the ballisticity

condition (T&#947; )

|l, and condition (T &#8242; )|l de&#64257;ned as the ful&#64257;llment of (T&#947; )|l for each

&#947;

&#8712; (0, 1). He proved that (T &#8242; ) implies ballisticity in the corresponding direction,

and showed that for each &#947;

&#8712; (0, 5, 1), (T&#947; ) implies (T &#8242; ). It is believed that for

each &#947;

&#8712; (0, 1), (T&#947; ) implies (T &#8242; ). We prove that for &#947; &#8712; (&#947;d , 1), (T )&#947; is equivalent

to (T &#8242; ), where for d

&#8805; 4, &#947;d = 0 while for d = 2, 3 we have &#947;d &#8712; (0.366, 0.388).

The case d

&#8805; 4 uses heavily a recent multiscale renormalization method developed

by Noam Berger. This talk is based on joint works with Alexander Drewitz from

ETH Z

urich.

l

# Fluctuations of ground state energy in Anderson model with Bernoulli potential

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Energy of the finite-volume ground state of a random Schroedinger operator is studied in the limit as the volume increases. We relate its fluctuations to a classical probability problem---extreme statistics of IID random variables---and describe the detailed behavior of its distribution. Surprisingly, the distributions do not converge---presence of two scales in the system leads to a chaotic volume dependence. A possible application to a sharp estimate of the Lifshits tail will be mentioned. The work presented is done jointly with Michael Bishop.