Lorentz Center - Continuous and Discrete Random Spatial Processes from 20 Apr 2004 through 29 Apr 2004
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### Continuous and Discrete Random Spatial Processesfrom 20 Apr 2004 through 29 Apr 2004

The super-process limit of oriented percolation above 4+1

# The Super-Process Limit of Oriented Percolation Above 4+1

Dimensions

Remco van der Hofstad, Eindhoven University of Technology

We consider oriented bond percolation on $\Zd \times \N$, at the critical occupation density $p_c$, for $d>4$.  The model is a spread-out'' model having long range parameterised by $L$.  We consider configurations in which the cluster of the origin survives to time $n$, and scale space by $n^{1/2}$.  We prove that for $L$ sufficiently large all the moment measures converge, as $n \to \infty$, to those of the canonical measure of super-Brownian motion.  This extends a previous result of Nguyen and Yang, who proved Gaussian behaviour for the critical two-point function, to all $r$-point functions ($r \geq 2$), and provides an example where an interacting model converges to super-Brownian motion. We use lace expansion methods for the two-point function, and prove convergence of the expansion using a general inductive method that we developed in a previous paper.  I will also describe some extension to the contact process, and mention several related (open) problems. This is joint work with Gordon Slade, Frank den Hollander and Akira Sakai.

## The Full Scaling Limit of 2D Critical Percolation

Federico Camia, EURANDOM, Eindhoven

Substantial progress has been made in recent years on the 2D critical percolation scaling limit and its conformal invariance properties. In particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter 6) was, in the work of Schramm and of Smirnov, indentified as the scaling limit of the critical percolation “exploration express.” In joint work with Chuck Newman, we use that and other results to construct what we argue is the full scaling limit of the collection of all closed contours surrounding the critical percolation clusters on the 2D triangular lattice. This random process or gas of continuum nonsimple loops in the plane is constructed inductively by repeated use of chordal SLE6. These loops do not cross but do touch each other – indeed, any two loops are connected by a finite “path” of touching loops.

## The Limit Behaviour of the Bak-Sneppen Model

Ronald Meester, Vrije Universiteit, Amsterdam

Abstract: Consider N points evenly spaced on a circle. At time 0, each point receives a so called fitness, a (uniform) random number between 0 and 1, independently of each other. The dynamics is as follows: (1) choose the point with the lowest fitness; (2) replace its fitness, together with the fitnesses of its two neighbours, by 3 new independent random numbers; repeat. Simulations by phycisists suggest that in stationarity, for large N, the fitnesses are uniformly distributed above a certain threshold b. Furthermore, they claim that the model shows self-organised critical behaviour in the sense that so called avalanches below this threshold have power law behaviour.  In this lecture, we make rigorous progress towards both these claims. We also discuss the so called gap-equation, which is supposed to describe how fast the system organises itself into the stationary state.

## Scaling Limit Processes in 2 and 1+1 Dimensions

Charles Newman, Courant Institute, NYU

Abstract: There are a number of striking continuum processes that arise as the scaling limits of fairly simple lattice systems in two spatial dimensions or in one space plus one time dimension: e.g., random walks with random rates, coalescing random walks, and critical 2D percolation. We will discuss some topics involving these continuum processes -- e.g., the characterization of the continuum nonsimple loop process (joint with F. Camia) and the relevance of the Brownian web for nucleation processes (joint with L.R. Fontes, M. Isopi, and K. Ravishankar).

How the Ising Crystal Grows?

Senya Shlosman, CPT CNRS, Marseille

I will discuss the question about how flat the flat facets of the canonical Ising model (random) crystal are. It turns out that they are not as flat as flat can be, and that monolayers of extra particles are forming on them. I will discuss various conjectures about this subject and explain the proofs of some of them for the case of SOS model and for the canonical Ising model with plus/minus boundary conditions.

Bond Percolation, the Raise and Peel Model and Alternating-Sign Matrices

Jan de Gier, University of Melbourne

Following a remarkable observation by Razumov and Stroganov in 2001, an intriguing connection has been conjectured between the square lattice O(n=1) model and combinatorial objects alled alternating-sign matrices (ASMs). This connection has resulted in many new exact results for the O(n=1) model, specifically for finite system sizes, which subsequently can be used for rigorous study of its scaling limit. However, a major problem, or rather challenge, is that while there being overwhelming evidence almost none of the results have been proved. Some of those that have been proved concern an interesting boundary dominated spatial structure of ASMs. Accepting the results, they have interesting physical applications. For example, it is well known that the (n=1) model is a model for bond ercolation, but I will show that it can also be interpreted as a stochastic model for interface growth. In addition, being a critical statistical mechanical model, it provides an example of a dynamical system with a spectrum described by conformal field theory. It is an open question how the conformal invariance appears on the level of the alternating-sign matrices.

## O(n) Conformal Field Theory, Calogero Models, and Possible Generalisations of SLE

John. L. Cardy, Oxford University

It was conjectured 20 years ago that points where the random curves of the O(n) model intersect the boundary of a domain correspond to insertions of boundary operators in the corresponding conformal field theory, whose correlators satisfy linear second-order differential equations. For single curves, this is nowadays understood in terms of the SLE description. We show that a generalisation to N curves leads to differential operators of the Calogero type. Their spectrum gives known as well as new bulk scaling dimensions of the O(n) model. The stochastic interpretation is a multi-particle generalisation of SLE in which the driving terms are governed by Dyson's Brownian motion. We also speculate on a way of counting self-avoiding loops using SLE in the full plane.

## A Phase Transition for a Model for the Spread of an Infection

Harry Kesten, Cornell University. Ithaca, New York

We consider a system which contains two types of particles, called A and B-particles. The A-particles are interpreted as healthy particles and the B-particles as infected ones. All particles perform independent, continuous time simple random walks on Z^d. The only interaction is that when an A-particle jumps onto a B-particle, or vice versa,  then the A-particle changes to a B-particle. We start with N(x,0) A-particles at x at time 0, where the N(x,0) are i.i.d. Poisson variables, and with finitely many B-particles at time 0.

First we shall discuss a shape theorem for the set of sites visited by an infected particle before time t, as t tends to infinity. We then extend the model and allow the infected particles to become healthy again at a fixed rate \lambda. That is, any B-particle turns independently  of everything else into an A-particle, at rate \lambda. We discuss a proof of a conjecture by R. Meester. For small \lambda the infection survives with positive probability, but for large \lambda the infection dies out almost surely.

## Random Walks on Percolation Clusters

Martin T. Barlow, UBC, Vancouver, Canada

This talk will discuss the SRW on percolation clusters. The first case is supercritical ($p>p_c$) bond percolation in Z^d$. Here one can obtain Aronsen type bounds on the transition probabilities, using analytic methods based on ideas of Nash. For the critical case ($p=p_c\$) one needs to study the incipient infinite cluster (IIC). The easiest situation is the IIC on trees - where the methods described above lead to an alternative approach to results of Kesten (1986). (This case is joint work with T. Kumagai).

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