# Interacting Particle Systems and Percolation

## Talks

Talks will be 50 mn long.

The TASEP speed process.

Authors : Amir, Angel, Valko.

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.

A microscopic concavity property and t1/3 scaling of current fluctuations in particle systems I.

Authors : Marton Balázs (Budapest), Julia Komjathy (Budapest), Timo Seppäläinen (Madison).

Abstract : One dimensional asymmetric particle systems with one conserved quantity can produce t1/3-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 t1/3 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.

Trap models for random walks on trees.

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.

Growth of random surfaces.

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.

Patterns formed by growing sandpiles.

Authors : D. Dhar, T. Sadhu, and S. Chandra.

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.

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.

Matrix representation of the stationary measure for the multispecies TASEP.

Authors : Martin R. Evans, Pablo A. Ferrari, Kirone Mallick.

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

Hydrodynamic limit of some multi-type exclusion processes via functional integration.

Authors : Guy Fayolle (Rocquencourt), Cyril Furtlehner (Saclay).

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.

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 Airy1 or the Airy2 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/0806.3410>].  A second aspect are the height-height correlations at different times. The space-time turns out to be non-trivially fibred, with some space-time curves with slow decorrelations [arXiv:0806.1350 <http://arxiv.org/abs/0806.1350 <http://arxiv.org/abs/0804.3035>>].

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.

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

tρ = (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.

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 μx(·) of the corresponding infinite system with translation invariant dynamics. This measure μx 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 μx = k(x,ρ) νρ dρ,  with νρ 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 νρ entering the decomposition of μx. 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.

The Symmetric Exclusion Process: Correlation Inequalities and Applications.

Authors : J. Borcea, P. Branden, T. M. Liggett.

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?

Multiclass queues and interchangeability.

Authors : Pablo Ferrari, James Martin, Balaji Prabhakar.

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.

Scaling Limit of the One-Dimensional Stochastic Potts Model.

Authors : C. Newman, K. Ravishankar and E. Schertzer.

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.

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.

A microscopic concavity property and t1/3 scaling of current fluctuations in particle systems II.

Authors : Marton Balázs (Budapest), Julia Komjathy(Budapest), Timo Seppäläinen (Madison).

Abstract : This talk discusses aspects of the proof of order t1/3 for current fluctuations in the asymmetric simple exclusion process and a class of zero range processes.

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.

Dynamical sensitivity of the infinite cluster in critical percolation.

Authors : Yuval Peres, Oded Schramm, Jeff Steif.

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 nα, 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.

Diffusive bounds for some self-interacting random walks with long memory in three and more dimensions.

Authors : Illes Horvath, Bálint Tóth, Bálint Veto.

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

Authors : Patricia Gonçalves, Claudio Landim and Cristina Toninelli.

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.

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.

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.

Passage time moments for the mixed voter-exclusion model.

Authors : Iain MacPhee, Mikhail Menshikov, Stas Volvok, Andrew Wade.

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.

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.

Survival time of random walk in random environment among soft obstacles.

Authors : Nina Gantert, Serguei Popov and Marina Vachkovskaia.

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).

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.

On shape stability for a storage model.

Authors : M.V. Menshikov, V.V. Sisko, V. Vachkovskaia.

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.

Survival of contact processes on the hierarchical group.

Authors : Siva R. Athreya and Jan M. Swart.

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.

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.

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains.

Authors : Mikhail Menshikov, Marina Vachkovskaia and Andrew Wade.

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
Tel : (33) 1 44 27 67 75                   Fax : (33) 1 44 07 09 37

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Last update : November 6, 2008.