YEP (Young European Probabilists)
Workshop on "Conformal invariance, scaling limits and percolation"
EURANDOM, Eindhoven, The Netherlands
March 29  to April 2, 2004

ABSTRACTS

Omer Angel, Université Paris-Sud

Scaling of an exploration process in random planar maps

Critical site percolation on a random planar triangulation is studied by using an exploration process for the percolation interfaces. This reduces various questions regarding percolation on these maps to questions involving simple Markov chains. In the simplest cases these Markov chains have scaling limits which are stable processes, and so exact parallels to Cardy's formula may be deduced. The technique enables consideration of other parameters of percolaiton, as well as other critical models.

Vincent Beffara, UMPA - ENS-Lyon

Schramm-Loewner Evolutions and scaling limits

The aim of these lectures is twofold:

* First, give a (mostly self-contained) introduction to SLE processes, which were introduced recently by Schramm as candidates to be the scaling limits of various critical two-dimensional discrete models; convergence to SLE has indeed been obtained in a few cases, which I will describe. (This should occupy the first 4 hours.)

* Second, I will present in some detail Smirnov's proof of Cardy's formula and convergence of the critical site-percolation cluster boundaries to SLE curves in the case of the triangular lattice. (Last 2 hours.)

Rob van den Berg, CWI

Inequalities in percolation and related fields

Inequalities, in particular so-called correlation inequalities, are important tools in probability theory. I will discuss some relatively old as well as some recent inequalities, give some intuitive background, and show how they are (or might be) used in percolation theory and related fields.

Cedric Boutillier, Université Paris-Sud

Dimer models : Scaling limit of edge densities

Consider the honeycomb lattice with edges weighted according to their direction. A dimer configuration is a subset of edges (dimers) such that every vertex of the lattice is incident with exactly one edge in this subset. A Gibbs measure can be explicitly constructed on these configurations, such that in some sense the probability of a configuration is proportional to the weights of the edges it contains. To every type of edge is associated a natural random field which gives the (random) number of edges of this type present in an arbitrary domain. We show that, once properly rescaled, this random field converges in distribution to a gaussian field, which is a derivative of the gaussian massless free field.

Federico Camia, EURANDOM

The Full Scaling Limit of 2D Critical Percolation

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, identified as the scaling limit of the critical percolation "exploration process." 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.

Per Hallberg, KTH, Sweden

"New" invariant Gibbs measures for the Potts model

Frank den Hollander, EURANDOM

The incipient infinite cluster for oriented percolation above 4 dimensions

Antal Járai, CWI

The incipient infinite cluster in two- and high-dimensional percolation

These lectures will address various aspects of large critical percolation clusters. Two different perspectives exist in studying asymptotic properties of large clusters: lattice scale view and the scaling limit. We will discuss the first of these. For the lattice scale viewpoint, one conditions a critical cluster to be large for example by requiring it to have radius larger than n, and takes the limit as n goes to infinity. This way an infinite cluster is obtained on the discrete lattice, called the incipient infinite cluster (IIC). The IIC captures what a large cluster looks like in a neighbourhood of one of its sites. It provides a natural setting to study problems like the asymptotic behaviour of random walk on a large critical cluster. The study of so called invasion percolation is also closely related to the IIC.

The approximate plan of the lectures is:

1. Motivation and Kesten's construction of the IIC in 2D

2. Various alternative constructions in 2D

3. The IIC in high dimensions

4. Random walk on critical branching process trees and the IIC

Ronald Meester, Vrije Universiteit

Fractal percolation versus classical percolation

I will introduce the fractal percolation process, and discuss some similarities and differences between classical discrete percolation on a lattice and this fractal percolation process. In particular, I will dicuss the existence of a phase transition, behaviour at the critical point, uniqueness of the infinite cluster, continuity of the percolation function, and connections between fractal and classical percolation.

Reda Messikh, Université Paris-Sud

Block renormalization for 2d FK-percolation

In this talk I will present an adaptation of the block renormalization of Pisztora to the planar case and show how it gives surface order large deviations estimates for some typical block events.

Bernard Nienhuis, University of Amsterdam

Critical lattice percolation on the strip and cylinder

In the last few years strong links have emerged between the bond percolation model on the square lattice and combinatorial problems like plane partitions and alternating sign matrices. So far these observed connections have retained the status of conjectures; no proof is available. Guided in part by the many theorems and conjectures that exist on the number of plane partitions and alternating sign matrices, we have formulated conjectures concerning the probability of events in percolation on the half infinite strip with finite width and the half cylinder with finite circumference. We will present a number of these conjectures. The scaling behavior of correlation functions in percolation have been known for a long time, but no or very few results concerning finite distances or finite geometry. By combining two half infinite cylinders correlations can be found on the entire cylinder. These conjectures on finite geometries permit a scaling limit to be taken from which universal quantities can be found. We find support for a continuous spectrum of critical exponents. These results not only fill in the gap between the scaling limit and the local details of the lattice model, but they lead to new questions and new knowledge about the percolation problem.

Frank Redig, Eindhoven University of Technology
Joint work with M Abadi, J.R. Chazottes, E. Verbitskiy and R. van der Hofstad.

Exponential laws for rare events in Gibbsian random fields

I will discuss exponential laws with precise error bounds for the occurrence, repetition and matching of patterns in Gibbsian random fields. Both high temperature regime and low temperature regimes will be discussed. Finally, we give applications to the asymptotics of maximal sub- and supercritical percolation clusters.

Akira Sakai, EURANDOM

Tomasz Schreiber, Nicolaus Copernicus University

Gibbsian modifications of polygonal Markov fields in the plane: phase transitions and random dynamics

Polygonal Markov fields, as introduced by Arak and Surgailis, are a particular class of continuum non-intersecting polygonal contour ensembles in the plane. The infinite-temperature free field admits a dynamic representation in terms of equilibrium evolution of one-dimensional particle systems tracing polygonal boundaries in two-dimensional space-time. In the present talk our interest is focused on finite-temperature Gibbsian modifications of such fields, with the Hamiltonian given by the total edge length. These continuum models, exhibiting translational and rotational symmetries, have recently been proven to share certain properties of the lattice Ising model, including the first order phase transition. We construct a family of random dynamics on contour collections which leave invariant the distributions of the considered polygonal fields. Appropriate infinite volume extensions of these dynamics allow us to characterise the stationary low temperature phases with tempered edge density and to obtain exponential mixing for the underlying contour ensembles. Further, we argue that with the stationarity and tempered edge density requirements lifted we get an infinite number of distinct phases breaking the translational and rotational symmetries and exhibiting untempered edge density growth. Some insight into the nature of these phases can be gained due to the fact that some polygonal fields arise as scaling limits of appropriate lattice models whose convenient feature is that they can be studied via an FK-type representation, lacking the FKG property though.

Stas Volkov, University of Bristol
Joint work with Codina Cotar

Some news from the lilypond

We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points Xi is fixed. At time zero simultaneously at each Xi a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle.

We study the expansions of this model in the following three directions. We study:  (1) a one-sided growth model with