Titles and Abstracts
________________________________________________________________________________
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.
________________________________________________________________________________
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.
________________________________________________________________________________
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.
-------------------------------------------------------------------------
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.
______________________________________________________________________________
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.
______________________________________________________________________________
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.
_______________________________________________________________________________
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.
__________________________________________________________
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)
________________________________________________________________________________
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.)
________________________________________________________________________________
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.
________________________________________________________________________________
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),
_______________________________________________________________________________
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.
-----------------------------------------------------
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.
________________________________________________________________________________
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.
_______________________________________________________________________________
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.
________________________________________________________________________________
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).
________________________________________________________________________________
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).
________________________________________________________________________________
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.