# Interacting Particle Systems and Percolation

## October 27th – 31th, 2008

## Talks

Talks will be 50 mn long.

**Omer Angel (Toronto): **
The TASEP speed process.

*Abstract : *
Start a multiple class asymmetric exclusion process on Z with a class-k
particle at position k (extending the notion of second class particles). It
follows from well known results, that each particle has an asymptotic
average speed U_k that is uniformly distributed in [-1,1]. We calculate the joint distribution of speeds. In particular we
prove that U_0 {<,=,>} U_1 with respective probability {1/3,1/6,1/2}. We
also describe the partition of the particles into infinite classes of
particles with equal speeds.

**Marton Balázs (Budapest): **
A microscopic concavity
property and t^{1/3}
scaling of current fluctuations in particle
systems I.

*Abstract : *
One dimensional asymmetric particle systems with one conserved quantity
can produce t^{1/3}-order
dynamical current fluctuations. A key
property that gives this behavior seems to be concavity or convexity of
the conservation law from the hydrodynamic procedure.

Within a class of systems that includes many well-known examples, I
will explain a microscopic version of that concavity or convexity
property, formulated at the particle level. Given a model with that
property I will outline the proof of the t^{1/3}
scaling.

Though the argument is robust and should be universal, the required
property seems to be too restrictive at the moment. Questions on how to
strengthen our present proof so that weaker assumptions suffice will
also be discussed.

In a later talk Timo Seppäläinen will give some ideas about the
couplings that can be used to verify the microscopic
convexity/concavity property. He will also give the few examples
(asymmetric
exclusion and zero range-type processes with exponential jump rates)
for which we could verify it and thus complete the program.

**Gérard Ben
Arous (New York): **
Trap models for random walks on trees.

**Rob van den Berg (Amsterdam): **
Approximate zero-one laws and sharp percolation transitions.

*Abstract : *
One of the most well-known classical results for site percolation on
the square lattice is that, for all values of the parameter p (except
its critical value) the following holds: Either a.s. there is an
infinite open cluster or a.s. there is an infinite closed `star'
cluster. This result is closely related to the percolation transition
being sharp.

We show how this result can be extended to a large class of 2D
percolation models, including 2D
Ising percolation (giving an
alternative proof of a 1993 result by Higuchi). I will also discuss
some open problems to which this research gave rise.

**Alexei
Borodin (Pasadena): **
Growth of random surfaces.

**Bernard
Derrida (Paris): **
Universal fluctuations of diffusive systems.

*Abstract : *
The first cumulants of the current of the symmetric
exclusion process for a ring geometry can be computed
exactly. For large system size, they take a scaling form
which can be understood by an analysis of the Bethe
ansatz equations. This scaling form can be recovered by a
theory based on fluctuating hydrodynamics, and it remains
valid for general diffusive systems.

**Deepak Dhar (Mumbai): **
Patterns formed by growing sandpiles.

*Abstract : *
Adding grains at a single site on a flat substrate in simple sandpile
models produces beautiful complex patterns, from the model's
simple deterministic local non-linear evolution rules. We have studied
in detail the pattern produced by adding grains in abelian sandpile
model on a two-dimensional directed F-lattice (each site has two
arrows directed inward and two outward), starting with a periodic
background with half the sites occupied.
The diameter of the pattern formed increases with the the
number of grains added N
as √N.
The pattern produced has a
countable infinity of patches, whith patches of density 1/2 alternating
with patches of density 1. We are able to characterize the asymptotic
pattern exactly, and determine the position and shape of
different patches.

**Rick
Durrett (Ithaca): **
Particle Systems on Random
Graphs.

*Abstract : *
Random graphs provide an
interesting new setting to investigate particle
systems previously studied on the d-dimensional integer lattice. In
this
talk we will discuss two examples, both joint work with my current
students. In the first we consider the contact process on a power law
random graph and show that despite mean field calculations the critical
value is always 0 for any power. In the second we will consider a model
for the spread of gypsy moths which alternates contact process growth
and
epidemics. On the random three regular graph and on the torus we find
chaotic behavior of the densities.

**Martin
R. Evans (Edinburgh): **
Matrix representation of the stationary measure for the multispecies
TASEP.

*Abstract : *
We consider a multispecies totally asymmetric exclusion process on the
ring ZL. The stationary state of the model with two species of particle
---first and second class particles where both species of particles hop
forwards and exchange with holes but the second class particles are
overtaken and interchange positions with the first class particles---has
previously been solved using the matrix product formulation by Derrida,
Janowsky, Lebowitz and Speer
1993. Also, probabilistic constructions of the steady state have
been made by Ferrari, Fontes and Kohayakawa 1994, Angel 2006, Ferrari
and
Martin 2007, the latter introducing a queueing system interpretation of the
stationary state. In this talk we demonstrate how the matrix product
formulation and the queueing interpretation are equivalent. Further we
show the stationary state for a system with any number of species of
particles may be written in the matrix product formulation in a
hierarchical fashion.

Reference: Martin R. Evans, Pablo A. Ferrari, Kirone Mallick, Matrix
representation of the stationary measure for the multispecies TASEP,
arXiv:0807.0327

**Guy
Fayolle (Rocquencourt): **
Hydrodynamic limit of some multi-type exclusion processes via
functional integration.

*Abstract : *
In statistical physics, interplay between discrete and
continuous
description is an important recurrent question. As for
reaction-diffusion systems, this amounts to studying fluid or
hydrodynamic limits. In particular, some classes of models dealing
with
the dynamics of discrete curves subject to
stochastic deformations (or
reactions with regard to polymers or biology) can be rephrased in
terms
of interacting exclusion processes. In order to derive hydrodynamic
equations after convenient scalings, a seemingly new general
functional
approach is proposed, illustrated by some classical models (ASEP,
ABC,
etc).The method resorts to variational calculus and functional
integration.

**Patrik Ferrari (Berlin): **
Limit processes in KPZ growth.

*Abstract : *
We consider the class of stochastic growth models in the
Kardar-Parisi-Zhang (KPZ) universality class. In 1+1 dimensions, for
large growth time t, the limit process describing the surface is the
Airy_{1} or the Airy_{2}
process, depending on the curvature of the limit
shape. The decay of the correlations are however very different
(superexponentially vs. polynomial) [arXiv:0806.3410 <http://arxiv.org/abs/0806.

**Jozsef
Fritz (Budapest): **
Microscopic derivation of isentropic
elasticity.

*Abstract : *
Hyperbolic scaling limit of the anharmonic chain with conservative,
Ginzburg-Landau type noise is investigated. The interaction potential is a
bounded perturbation of the quadratic one; total momentum and deformation are
preserved at the microscopic level. Hydrodynamic limit of the model results in
the physically relevant p-system, that is the nonlinear sound equition in one
space dimension. In a smooth regime it is sufficient to add noise only to
velocities, but in the presence of shocks scaling-dependent noise is added to
both components, and the method of compensated compactness is used.

**Claudio
Landim (Rouen and Rio): **
Hydrodynamic
limit of gradient exclusion processes with conductances.

*Abstract : *
Fix a strictly increasing right continuous with left limits
function W: R→
R and a smooth function Φ: [l,r] →
R, defined on some interval [l,r] of R,
such that 0<b ≤
Φ ≤ b^{-1}.
We prove that the evolution, on the diffusive scale, of the
empirical density of exclusion processes, with conductances
given by W,
is described by the weak solutions of the non-linear
differential equation

∂_{t}ρ = (d/dx)(d/dW)Φ(ρ).

We derive some properties of the operator (d/dx)(d/dW) and prove uniqueness of weak solutions of the previous non-linear differential equation.

**Joel
L. Lebowitz (Rutgers): **
Local and Global Structure of stationary states of macroscopic systems.

*Abstract : *
The microscopic structure
of a macroscopic system in a steady state is described locally, i.e. at
a suitably scaled macroscopic point x, by a time
invariant measure
μ_{x}(·)
of the corresponding infinite system with
translation invariant dynamics. This measure μ_{x} may be extremal,
with good decay of correlations, or a superposition of extremal
measures, with weights depending on x (and possibly
even on the way one
scales). In the latter case a possible scenario would be to
have μ_{x}
= ∫
k(x,ρ) ν^{ρ}
dρ,
with ν^{ρ}
a
translation invariant extremal measure of the infinite system with
density ρ.

We expect that the microscopic configuration in the vicinity of x will
fluctuate, over sufficiently long times, between configurations typical
for the different ν^{ρ}
entering the decomposition of μ_{x}.
One expects (and proves in some cases) that there are equations,
deterministic or stochastic, describing the evolution, on the
appropriate hydrodynamic scale, of the macroscopic state of the system.
As the parameters of the dynamics are changed or the globally conserved
densities are varied, the system may undergo phase transitions which
are reflected in the k(x,ρ). I
will illustrate the above by some
exact results for 1D lattice systems with two types of particles (plus
holes) evolving according to variants of the simple asymmetric
exclusion process, in open or closed systems. Somewhat surprisingly,
the spatially asymmetric local dynamics satisfy (in some cases)
detailed balance with respect to a global Gibbs measure with long range
pair interactions.

**Thomas M.
Liggett (Los Angeles): **
The Symmetric Exclusion Process: Correlation
Inequalities and Applications.

*Abstract : *
In recent work with J. Borcea and P. Branden, we obtained
strong negative correlation properties for the symmetric exclusion
process. I will explain how they come about, and apply them
to solve a problem posed by R. Pemantle a few years ago. Here it is:
Start a one dimensional process with all negative sites
occupied, and all positive sites vacant. Is there a CLT for
the number of particles to the right of the origin at time t?

**James
Martin (Oxford): **
Multiclass queues and interchangeability.

*Abstract : *
Consider a "./M./1 queue" - that is, a queue with a single exponential
server. Burke's theorem tells us (among other things) that a Poisson process
is a "fixed point" for this queue; if the arrivals are a Poisson process,
then so are the departures. I'll talk about extensions to priority queues
with two or more classes of customers. The fixed points can be related to
equilibria of multiclass exclusion processes. I'll emphasise the role played
by ideas of interchangeability of queues. There are interesting clustering
phenomena, and some new solutions for directed percolation problems.

**Chuck
Newman (New York): **
Scaling Limit of the One-Dimensional Stochastic Potts Model.

*Abstract : *
The scaling
limits of one-dimensional stochastic Ising or Potts models at zero
temperature can be easily expressed in terms of the Brownian web (the
scaling limit of coalescing random walks or equivalently of the
one-dimensional voter model). If one considers Ising (but not Potts)
models at nonzero temperature (appropriately scaled to zero) or voter
models with noise, then the scaling limit has been related (in earlier
joint work with L. R. G. Fontes, M. Isopi and Ravishankar) to a
Brownian web with a relatively simple Poissonian marking. In the case
of Potts models at nonzero temperature, the scaling limit involves a
more complicated type of marking which is related in spirit to the type
of marking proposed earlier (jointly with F. Camia and Fontes)
for scaling limits of near-critical and of dynamical two-dimensional
critical percolation.

**Gunter M.
Schütz (Jülich): **
Exact solution of the Bernoulli matching model of sequence
alignment.

*Abstract : *
We consider the Bernoulli matching model of sequence
alignment. We map this problem to the discrete-time totally asymmetric
exclusion process with backward sequential update and step function
initial condition. Using earlier results obtained from Bethe ansatz
allows us to derive the exact distribution of the length of the longest
common subsequence of two sequences of finite lengths X,Y. Asymptotic
analysis adapted from Johansson's work allows us to derive the
thermodynamic limit directly from the finite-size result.

**Timo Seppäläinen (Madison): **
A microscopic concavity property and t^{1/3}
scaling of current
fluctuations in particle systems II.

*Abstract : *
This talk discusses aspects of the proof of order t^{1/3}
for current
fluctuations in the asymmetric simple exclusion process and a class of
zero range processes.

**Sunder
Sethuraman (Ames): **
Tagged particle asymptotics in certain zero-range and exclusion
systems.

*Abstract : *
We discuss, for a tagged particle in diffusive scale in
one dimension, two results which connect to the underlying
hydrodynamics: Fluctuations in mean-zero zero-range models, and large
deviations in the nearest-neighbor symmetric exclusion system.

**Jeff Steif (Göteborg): **
Dynamical sensitivity of
the infinite cluster in critical percolation.

*Abstract : *
We look at dynamical percolation in the case where percolation occurs
at criticality. For spherically symmetric trees, if the expected number of
vertices at the n-th
level connecting to the
root is of the order n(log
n)^{α},
then if α
> 2,
there
are no exceptional times of nonpercolation while if α ∈ (1,2),
there are such exceptional times. (An older result of R. Lyons tells us
that percolation occurs at a
fixed time if and only if α >1.) It turns
out that within the regime where there are no exceptional
times, there is another type of ``phase transition'' in the behavior of
the process. If the expected number of vertices at the n-th level
connecting to the root is of the form n^{α},
then if α
> 2, the number of connected components of the
set of times in [0,1] at which the root is not percolating is
finite a.s. while if α ∈ (1,2),
then the number of such components is
infinite
with positive probability.

**Bálint Tóth (Budapest): **
Diffusive bounds for some self-interacting random walks with long
memory in three and more dimensions.

*Abstract : *
First I survey older conjectures and results regarding the so-called
'true' self avoiding random walk. Then I present a fresh result
regarding diffusive bound for a variant of the three dimensional case.

## Posters

*Abstract : *
We study a conservative particle system with degenerate rates, namely
with nearest neighbor exchange rates which vanish for some
configurations. Due to this degeneracy the hyperplanes with fixed
number of particles can be decomposed into an irreducible set of
configurations plus isolated configurations that do not evolve under
dynamics.

We show that, for initial profiles smooth enough and bounded away from
zero and one, under the diffusive scaling, the macroscopic density
evolves according to the porous medium equation.

Then we prove the same result for more general profiles for a slightly
perturbed microscopic dynamics: we add jumps of the Symmetric Simple
Exclusion which remove the degeneracy of rates and are
properly slowed
down in order not to change the macroscopic behavior.

The equilibrium fluctuations and the magnitude of the spectral gap for
this perturbed model are also obtained.

**Júlia Komjáthy (Budapest): **
Order of current variance
and diffusivity in the rate one totally asymmetric zero range
process.

*Abstract : *
In the poster we present a simplified proof of the $t^{2/3}$-order
of the integrated particle current across the characteristics in
the rate one totally asymmetric zero range process. We use
probabilistic methods to couple together stationary processes at
different densities and consider the behavior of the second class
particles between them. We perform a delicate analysis of a single
second particle on one of the processes in the environment presented
by the two coupled processes. This will lead to the proof of the
$t^{2/3}$-scaling. Via this method we will see the phenomenon of
microscopic concavity, i.e. a possible analog of the concavity of
the hydrodynamic flux at microscopic level, including that single
second class particles on coupled systems with different densities
can stay ordered. The method will be presented via pictures and will
aim to be as understandable as possible.

The poster is closely related to the talks of M. Balázs and T. Seppäläinen.

**Fabio Machado (São Paulo): **
Non-homogeneous random walks systems on Z.

*Abstract : *
We consider a random walk system on Z in which each active
particle performs an asymmetric simple random walk and activates
all inactive particles it encounters. The movement of an active
particle stops when it reaches a certain number of jumps without
activating any particle. We prove that if the process counts on
efficient particles (small probability of jumping to
the left) placed strategically on Z, the process might
survive, having active particles at any time with positive probability.
On the other hand, we may construct a process that dies
out eventually almost surely, even counting on efficient particles.
That is, we discuss what happens if particles are placed
initially very far away from each other or if their probability of jumping
to
the right tends to~1 but not fast enough.

**Iain MacPhee (Durham): **
Passage time moments for the mixed voter-exclusion model.

*Abstract : *
We consider a one-dimensional discrete-time interacting particle
system
that is a mixture of the symmetric voter model and asymmetric exclusion
process (either transient or recurrent exclusion). Starting from a
configuration that is a finite perturbation of the Heaviside
configuration, we study the moments of the relaxation time of the
process. Our results include some progress towards a conjecture of
Belitsky et al. [Bernoulli 7 (2001) pp 119--144] that sufficiently
amount of asymmetric exclusion prevents the model being positive
recurrent. Moreover, we have results on the almost-sure evolution of
the size of the hybrid (disordered) region, via an application of
general criteria for obtaining almost-sure bounds for stochastic
processes. Several interesting open problems remain.

**Jose Carlos Simon de Miranda (São Paulo): **
Non ruin probability under time-varying risk process, premium and interest
rates.

*Abstract : *
We determine the infinite horizon non ruin probability, defined as a
function
of instantaneous reserve capital and time, for a risk model under time
dependent
interest rates where the risk process is a Poisson non homogeneous point
process
with time dependent distribution of marks, i.e. claim sizes, and the income
premium is also time dependent. The explicit form of this probability
function
is obtined in frequency domain. A limit model and its particular case, where
the
claims are time-varying exponentially distributed, are presented.

**Serguei Popov (São Paulo): **
Survival time of random walk in random environment among soft
obstacles.

*Abstract : *
We consider a Random Walk in Random Environment (RWRE) moving in a
i.i.d.
random field of obstacles. When the particle hits an obstacle, it
disappears with a positive probability. We obtain quenched and
annealed bounds on the tails of the survival time in the
general *d*-dimensional
case. We then consider a simplified one-dimensional model (where
transition probabilities and obstacles are independent and the RWRE
only
moves to neigbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the ``mixed" probability
measures (quenched with respect to the obstacles and annealed with
respect
to the transition probabilities and vice-versa) and give results for
tails
of the survival time with respect to these probability measures.
Further,
we apply the same methods to obtain bounds for the tails of the
hitting
times of Branching Random Walks in Random Environment (BRWRE).

**Pablo Martín Rodríguez (São Paulo): **
An upper bound for the critical probability of a long range
percolation model on trees.

*Abstract : *
We consider a simple long range percolation model on infinite
graphs and discuss sufficient conditions for phase transition. In addition,
we study the model on spherically symmetric trees and obtain an upper bound
for the critical probability. As a consequence we obtain the asymptotic
behavior of this critical parameter on homogeneous trees.

**Valentin Sisko (São Paulo): **
On shape stability for a storage model.

*Abstract : *
We consider stability of shape for a storage model on $n$
nodes. These nodes support ${\cal K}$ neighborhoods $S_i\subset \{1,
\ldots, n\}$ and items arrive at the $S_i$ as independent Poisson
streams with rates $\lambda_i$, $i=1$, $\ldots\,$, ${\cal K}$. Upon
arrival at $S_i$ an item is stored at node $j \in S_i$ where $j$ is
determined by some policy. Let $X_j(t)$ denote the number of items
stored at $j$ at time $t$ and let $X(t) = \bigl( X_1(t), \ldots\, ,
X_n(t) \bigr)$. Under natural conditions on the $\lambda_i$ we
exhibit simple local policies such that $X(t)$ is positive recurrent
(stable) in shape.

**Jan Swart (Prague): **
Survival of contact processes on the hierarchical group.

*Abstract : *
We consider contact processes on the hierarchical group, where sites infect
other sites with a rate depending on their hierarchical distance, and sites
become healthy with a fixed recovery rate. If the infection rates decay too
fast as a function of the hierarchical distance, then we show that the
critical recovery rate is zero. On the other hand, we derive sufficient
conditions saying how fast the infection rates can decay while the critical
recovery rate is still positive. Our proofs are based on a coupling argument
that compares contact processes on the hierarchical group with contact
processes on a renormalized lattice. For technical simplicity, our main
argument is carried out only for the hierarhical group with freedom two.

**Jan Swart (Prague): **
The contact process seen from a typical infected site.

*Abstract : *
We consider contact processes on general Cayley graphs. It is shown that any
such contact process has a well-defined exponential growth rate, which can
be related to the configuration seen from a `typical' infected site at a
`typical' late time. Using this quantity, it is proved that on any
nonamenable Cayley graph, the critical contact process dies out.

**Marina Vachkovskaia (Campinas): **
Asymptotic behaviour of randomly reflecting billiards in unbounded
tubular domains.

*Abstract : *
We study stochastic
billiards in infinite planar domains with
curvilinear boundaries: that is, piecewise deterministic
motion with
randomness introduced via random reflections at the domain
boundary. Physical motivation for the process originates with ideal gas
models in
the Knudsen regime, with particles reflecting off
microscopically rough
surfaces. We classify the process into recurrent and
transient cases.
We also give almost-sure results on the long-term behaviour of the
location
of the particle, including a super-diffusive rate of escape in the
transient case. A key step in obtaining our results is to relate our
process to an instance of a one-dimensional stochastic process
with
asymptotically zero drift, for which we prove some new almost-sure
bounds.

**IHP - Centre Emile Borel - 11, rue Pierre et Marie Curie 75005 Paris - France**

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Last update : November 6, 2008.