Titles and Abstracts

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The two-type contact process

Enrique Andjel

Abstract: We study a one dimensional two-type contact process and give necessary

and sufficient conditions on the initial configuration for

both types to survive forever. These results are proved under the

assumption that the rates of propagation (and death) of the two types

are equal.

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Stability Criteria and Applications for Randomized Load Balancing Schemes

Maury Bramson

In this talk, we consider randomized load balancing schemes where an arriving

job joins the shortest of $d$ randomly chosen queues from among a pool of $n$

queues. Vvekenskaya, Dobrushin, and Karpelevich (1996) considered the case

with Poisson input and exponentially distributed service times, and derived an

explicit formula for the equilibrium distribution for fixed $d$ as $n$ goes to

infinity. Since its tail decays doubly exponentially fast, this distribution is

useful in various applications.

Relatively little work has been done for general service times or input. For

general service times, the behavior of the service rule at each queue will now

play a role in the behavior of the system. In particular, the question of

under which conditions the system is stable (i.e., its underlying Markov

process is positive recurrent) for fixed $n$ is no longer obvious. Ideally,

one would like to understand the limiting behavior of such equilibria (provided

they exist) as $n$ goes to infinity, as in the first paragraph.

Here, we discuss results that show that for fixed $n$ such systems are always

stable for the appropriate notion of traffic intensity. These results also

show that the associated equilibria are tight when restricted to a finite

number of queues and hence subsequential limits exist as $n$ goes to infinity.

It is anticipated that this behavior will provide a general framework for

examining the behavior of such limits under different service rules. In this

context, we discuss joint work with Y. Lu and B. Prabhakar on the limiting

behavior of the equilibria when service at each queue is given by the standard

first-in, first-out rule.

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Starting from T.M. Liggett's paper (1989) on $L_2$-exponential convergence

Mu-Fa Chen (Beijing Normal University)

Abstract: In his first paper on the $L_2$-exponential convergence (Ann. Probab.,

17:2, 1989), T.M. Liggett's studied the spectral gap for birth-death

processes

and then used as a tool to the problem for the attractive reversible nearest

particle systems. That paper was our starting point for studying the speed of

stability, to which the $L_2$-exponential convergence is a typical one. In this

talk, we survey some progress made in the past twenty years along this

direction.

(1) A diagram of various types of ergodicity and their explicit criteria

in dimension one.

(2) The non-ergodic case.

(3) The killing case.

The most part of the talk is concentrated on the birth-death processes.

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Invariant random graphs with prescribed iid degrees

Maria Deijfen

Models for generating random graphs with prescribed degree distribution

have been extensively studied the last few years. Most existing models for

this purpose however do not take spatial aspects into account, that is,

there is no metric defined on the vertex set. I will discuss spatial

versions of the problem. More precisely, given a degree distribution F and

a spatial vertex set - for instance Z^d or the points of a spatial Poisson

process - how should one go about to obtain a translation invariant random

graph on the given vertex set with degree distribution F? Which properties

do the resulting configurations have? I will describe some existing

results and a number of open problems.

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Ising model on complete graphs and trees

Jian Ding

The following picture for Glauber dynamics for the Ising model was

discovered by physicists and fine-tuned

by mathematicians. Denote by $\beta_c$ be the critical

inverse-temperature. In the subcritical regime

(namely, $\beta < \beta_c$,), the mixing time is fast and the convergence

is sharply concentrated. In the

critical regime ($\beta = \beta_c$), the inverse-gap is polynomial in the

cut-width of the underlying graph. In

the supercritical regime ($\beta > \beta_c$), the mixing is very slow due

to coexistence of phases. However, one

can eliminate the bottleneck between the multiple phases and reestablish

efficient mixing by running a restricted dynamics or imposing suitable

boundary conditions. We prove this picture completely for an underlying

geometry of a complete graph, and partly for regular trees (at criticality).

Joint work with Eyal Lubetzky and Yuval Peres.

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Voter Model Perturbations

Rick Durrett

The block construction has often been used to study coexistence in particle

systems with long range or rapid stirring. In this talk we will describe

another situation in which this method can be employed: for systems that

are small perturbations of the voter model. The motivating example is the

Neuhauser-Pacala competition model, but there are a number of other

interesting examples. The work described is joint with Ted Cox, Ed Perkins,

and Daniel Remenik.

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Multi class asymmetric processes and multitype customer queues

Pablo Ferrari

Multiclass processes appear naturally when the "basic coupling" of Liggett is

used for several marginals. Recently the construction of invariant measures for

some of those processes have been obtained. The invariant measure for the Hammersley

process, for instance, is the fixed point for a multitype ./M/1 queue.

I will review these and related results. Joint work with James Martin.

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Bootstrap percolation on regular trees has 2 phase transitions

L. Renato Fontes

We study the threshold theta bootstrap percolation model on the

regular tree with degree b+1, 2 leq theta leq b, and initial density

p. It is known that there exists a nontrivial critical value for p,

which we call p_f, such that a) for p>p_f, the final bootstrapped

configuration is fully occupied for almost every initial

configuration, and b) if p<p_f, then for almost every initial

configuration, the final bootstrapped configuration has density of

occupied vertices less than 1.

In this talk, we discuss the existence of a distinct critical value

for p, p_c, such that 0<p_c<p_f, with the following properties:

1) if p leq p_c, then for almost every initial configuration there is

no infinite cluster of occupied vertices in the final bootstrapped

configuration;

2) if p>p_c, then for almost every initial configuration there are

infinite clusters of occupied vertices in the final bootstrapped

configuration.

(joint work with Roberto Schonmann)

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Random-parity representation for the quantum Ising model

Geoffrey Grimmett

The quantum Ising model in $d$ dimensions may be mapped to a $(d+1)$-dimensional
`continuous' Ising model of classical type. This may be solved using an approach
developed by Aizenman and others under the name `random-current representation'.
We prove the sharpness of the phase transition, and establish two inequalities
for critical exponents. The value of the ground-state critical point may be
calculated rigorously in one dimension, and the corresponding transition is
continuous. (Joint work with Jakob Bj\"ornberg.)

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Signed Voter models on Graphs (joint with E. Andjel and G. Maillard)

Tom Mountford

We consider the signed voter model corresponding to random walks on graphs.

Unlike the classical voter model, it may well be the case that such models are

ergodic and a major question is to identify such models.

We address some questions raised by Gantert, Lowe and Steiff. In particular

they showed that ergodicty was strongly related to the property of

random walks

producing infinitely many "unsatisfied cycles" but that this was

not necessary for ergodicity to hold. We show that for random walks

on the integer lattice indeed the property is not necessary but in

dimension three it is.

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Limit Theorems for Shortest Remaining Processing Time Queues

Amber Puha (California
State University San Marcos)

A shortest remaining processing time (SRPT) queue is a single station queue to which a single class of jobs arrive each requiring a possibly random amount of processing time. Rather than serving jobs in a first-come-first-serve fashion, jobs are prioritized according to their current remaining service requirements (residual service times). The job with the shortest residual service time is the one that is processed. This is done with preemption so that if a job arrives that has a shorter service requirement than the residual service time of the job currently in service, then the job currently in service is placed on hold and the new job enters service. The SRPT service discipline is naturally of interest because it minimizes the queue length. An additional feature of the SRPT service discipline is that in order to obtain a Markovian state descriptor, one must track all of the residual service times. Hence, any Markovian state descriptor is necessarily infinite dimensional. In such settings, a recent trend in stochastic networks is to choose a measure-valued state descriptor to succinctly describe the system state. In the talk, a measure valued state descriptor for the SRPT queue will be introduced. Then two limit theorems that can be proved for this state descriptor will be described.

Joint work with Doug Down (McMaster University), H. Christian Gromoll (University of Virginia), and Lukasz Kruk (Maria Curie-Skladowska University),

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Couplings, Attractiveness and Hydrodynamics for

Conservative Particle Systems.

Ellen Saada

Abstract: This is a joint work with Thierry Gobron.

Attractiveness is a fundamental tool to study interacting

particle systems and the basic coupling construction is a

usual route to prove this property, as for instance in

simple exclusion. The derived Markovian coupled process

$(\xi_t,\zeta_t)_{t\geq 0}$ satisfies:

(A) if $\xi_0\leq\zeta_0$ (coordinate-wise), then

for all $t\geq 0$, $\xi_t\leq\zeta_t$ a.s.

In this work, we consider generalized misanthrope models

which are conservative particle systems on $\Z^d$ such that,

in each transition, $k$ particles may jump from a site $x$

to another site $y$, with $k\geq 1$. These models include

simple exclusion for which $k=1$, but, beyond that value,

the basic coupling construction is not possible and a more

refined one is required. We give necessary and sufficient

conditions on the rates to insure attractiveness; we

construct a Markovian coupled process which both satisfies

(A) and makes discrepancies between its two marginals

non-increasing. We determine the extremal invariant and

translation invariant probability measures under general

irreducibility conditions.

We apply our results to examples including a two-species

asymmetric exclusion process with charge conservation (for

which $k\le 2$) which arises from a Solid-on-Solid interface

dynamics, and a stick process (for which $k$ is unbounded) in

correspondence with a generalized discrete

Hammersley-Aldous-Diaconis model. We derive the hydrodynamic

limit of these two one-dimensional models.

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A waiting time problem arising from the study of multi-stage carcinogenesis

Jason Schweinsberg

ABSTRACT: We consider a model of a population of fixed size N in which each

individual gets replaced at rate one and each individual experiences a

mutation at rate u. We calculate the asymptotic distribution of the time

that it takes before there is an individual in the population with m

mutations. A variety of different behaviors are possible, depending on

how u changes with N. These results have applications to the problem of

determining the waiting time for regulatory sequences to appear and to

models of cancer development. This talk is partly based on joint work

with Rick Durrett and Deena Schmidt.

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Slow bond "demystified"

V. Sidoravicius

Consider one dimensional Totally Asymmetric Exclusion Process

(TASEP) on $\mathbb Z$, with the density of particles being 1/2,

and jump rates being 1 everywhere but the origin, where the jump rate is

different and equals to $1 -\lambda, \; 0 < \lambda < 1$.

One of the central questions for this model is to find out for which

values of $\lambda$ the current is affected on the far-right-side

of the system? This problem is known as the "slow bond problem", and the

search for the critical value $\lambda_c$ was one of main questions.

The model has several equivalent alternative representations as the

Polynuclear Growth Model (PNG) with columnar defect or as directed Last

Passage Percolation with reinforced diagonal. It also can be treated by

Random Matrices techniques considering it in the language of generalized

permutations.

In this talk I will address to this problem using Polynuclear Growth

language, and will show that $\lambda_c = 0$, which is in agreement

with the predictions by Janowski and Lebowitz (1991). The proof uses

Interacting Particle Systems approach and in some sense is "conceptual".

The talk based on joint works with V. Beffara, T. Sasamoto and M.E. Vares.

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The dynamical circle covering problem

Jeff Steif

Abstract: The classical circle covering problem introduced by Dvoretzky

is the following question: Suppose that I_1,I_2,I_3,... are intervals

of decreasing lengths l_1,l_2,l_3,... and that these intervals are

independently and uniformly distibuted on the circle of unit circumference. It

is a trivial consequence of Borel-Cantelli that any given point on the circle

will a.s. be covered by infinitely many of these intervals iff the sum of the

lengths is divergent, but will the whole circle be covered? Partial results on

this were obtained by Dvoretzky and many others. Finally, Shepp (1972) proved

the answer is a.s. yes iff \sum_{n=1}^\infty \frac{e^{l_1+\ldots+l_n}}{n^2} =

\infty. Assuming that l_n = c/n for a constant c, this implies that the whole

circle will be covered infinitely often iff c \geq 1 (a special case which was

known before Shepp).

We will consider a dynamical version of the problem where the

intervals after having been given initial random positions move

according to independent standard Brownian motions. Assume

that l_n = c/n for a constant c. Among other things we show that for c<2 a.s.

there are exceptional times when a given fixed point is covered only finitely

often, whereas this is not the case when c>=2. We also show that when c<3

there are a.s. exceptional times where the circle is not covered infinitely

often, whereas when c>=3 the whole circle is covered all the time. This is

joint work with Johan Jonasson.

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Stochastic flows of kernels in the Brownian net and the Brownian web

Rongfeng Sun

Abstract: Le Jan and Raimond developed a theory of stochastic flows of

kernels, showing that each consistent family of n-point motions gives rise

to a stochastic flow of kernel. As an example, they constructed a

special family of such flows where the underlying one-point motion is a

Brownian motion on R. Recently, Howitt and Warren introduced a much more

general class such flows on R, where the underlying n-point motions are

Brownian motions with sticky interactions upon collision. Stochastic flows

of kernels can be interpretated as random motion in a stationary

space-time random environment, where the environment satisfies certain

independent innovation properties. Here we give a graphical construction

of the underlying environment for the Howitt-Warren flow in terms of the

Brownian net (resp. the Brownian web), which loosely speaking consists of

a collection of branching-coalescing (resp. coalescing) Brownian

motions starting from every point in the space-time plane. Almost sure

path properties for the Howitt-Warren flow will also be derived.

This is based on joint work in progress with Jan M. Swart (UTIA, Prague)

and Emmanuel Schertzer (Columbia, New York).

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Randomized poly-nuclear growth with a columnar defect

M. Eulalia Vares

In this talk we examine a variant of the poly-nuclear growth model where

the level boundaries perform continuous-time, discrete-space random walks.

We examine how its asymptotic behavior is affected by the presence of a

columnar defect on the line, proving that there is a non-trivial phase

transition in the strength of the perturbation, above which the law of

large numbers for the height function is modified.

The talk is largely based on recent work in collaboration with Vincent

Beffara and Vladas Sidoravicius (to appear in Probability Theory and Related
Fields).

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Linear Stochastic Evolutions: Diffisive Behavior and Localization.

Nobuo YOSHIDA (Kyoto University)

We consider stochastic growth model on

the $d$-dimensional lattice. The time evolution is

described by successively multiplying i.i.d. random matrix

(Chapter IX of Prof. Liggett's book: ``Interacting Particle Systems'')

In continuous-time case, the model containes the Potlach model

and the binary contact path process.

Also in discrete-time case, the model describes

various interesting examples such as

oriented site/bond percolation, directed polymers in random environment,

time discretizations of binary contact path process. We first investigate

the regular/slow growth phase transition in terms of the growth rate of

the total populaiton. Then, we explain that

the regular/slow growth phase transition

is related to the delocalization/localization transition of the

spatial distribution of the population.