YEP 2004 (Young European Probabilists)
workshops

Workshop on
"Conformal invariance, scaling limits and percolation"
EURANDOM, Eindhoven, The Netherlands
March 29  to April 2, 2004

# 1. Scientific scope

In the past years, there has been substantial progress in the understanding of scaling limits for critical percolation models. These scaling limits describe the shape of large percolation clusters, such as their dimension and spatial extent. Descriptions of these scaling limits allow for the study of universality, meaning that the global properties of large critical clusters do not depend on the detailed local description of the models such as the underlying lattice.

Particular progress has been made in two dimensions using the notion of conformal invariance, and above the upper critical dimension six, using the lace expansion. We will now highlight the results in these two cases.

Two dimensions and the Stochastic/Schramm’s Loewner Evolution.

In a seminal paper [22], Oded Schramm proposed a new class of models, Stochastic Loewner Evolution (SLE), which are two-dimensional models that are conformally invariant. Conformal invariance of a stochastic process means that the law of the process remains unchanged when applying a conformal map to the process. This was important progress for the understanding of critical two-dimensional statistical mechanical models, since many are believed to have conformally invariant limits. The SLE processes are natural candidates for the laws of such scaling limits. Later this process was called Schramm-Loewner Evolution.

SLE can be obtained by substituting Brownian motion in the Loewner differential equation (see [18]). This family of differential equations, indexed by , is characterized by a driving function. Since the solution of the differential equation may explode in finite time, we obtain growing regions by investigating those  for which the solution has exploded before time . One can obtain SLE by taking as a driving function Brownian motion with variance . For each , this defines a conformally invariant process, and the precise characteristics of this process depend sensitively on the parameter . For instance, for , a simple curve is obtained, while for , the process is space-filling.

Originally, Schramm [22] used SLE to prove that the scaling limits of loop-erased random walk is SLE2 and that the scaling limit of uniform spanning trees is SLE. In a series of papers, Lawler, Schramm and Werner investigated properties of SLE, and showed that it can be used to compute the intersection exponents of Brownian motion [13,14,15].

For percolation, Schramm conjectured that the scaling limit of critical percolation is SLE. This scaling limit SLE would then confirm well-known predictions arising in the physics literature, such as the computation of critical exponents by Nienhuis [19,20], or the prediction of the scaling limits of crossing probabilities known as Cardy’s formula [3,4]. Crucial progress was made when Smirnov showed that two-dimensional site percolation on the triangular lattice is conformally invariant, thus showing that the scaling limit indeed is SLE [23]. This fundamental result has since been used to identify a number of critical exponents as predicted by Nienhuis, such as the one-arm exponent [16], and the critical exponent for subcritical cluster sizes [24].

The lace expansion for high-dimensional percolation models.

The lace expansion first appeared in 1985 in [2], where Brydges and Spencer used it to prove that in dimensions greater than 4 the end-point of weakly self-avoiding walk behaves diffusively. Weakly self-avoiding walk can be seen as simple random walk of a fixed length that is penalized by a factor  for every self-intersection. The lace expansion is a combinatorial resummation identity that relates (weakly) self-avoiding walks of length  to self-avoiding walks of shorter length multiplied with an expansion term having a certain diagrammatic description. This diagrammatic description can be used to bound the expansion term in terms of (weakly) self-avoiding walks again. The above method leads to a consistency requirement, and it can be shown that in dimension greater than 4 and sufficiently small interaction parameter , weakly self-avoiding walk behaves as Brownian motion.

Although Brydges and Spencer invented the lace expansion, the strongest results have been obtained by Hara and Slade. Slade [25,27] realized that also strictly self-avoiding walk, obtained when , can be handled using the lace expansion when the dimension is sufficiently large. Moreover, he also identified the scaling limit to be Brownian motion in [26]. Later, Hara and Slade realized that the method could be extended all the way down to the upper critical dimension. This lead to the celebrated result that self-avoiding walk in  or more dimensions has the same scaling as Brownian motion [8]. It is expected that this Brownian scaling does not hold in 3 or fewer dimensions, and that in dimension 4 there are logarithmic corrections to the diffusive scaling.

Moreover, around 1990, Hara and Slade [6,7,9] investigated different models where the lace expansion can be used. These other models are lattice trees and lattice animals, and percolation. Lattice trees are embedded trees in  containing no cycles, whereas lattice animals are embedded graphs in . The extension to percolation in [6] is complicated, as the lace expansion can only be performed using inclusion-exclusion, due to the fact that the interaction for percolation is more deeply hidden.

Initially, Hara and Slade only investigated so-called two-point functions. In the case of percolation, this is the probability that a point  is connected to the origin. Their results from 1990 suggest that we can think of geometric objects, such as the cluster of the origin and the lattice tree or animal above the respective critical dimension, as containing a simple random walk path starting at 0 and leading to , with branches or hairs originating along the random walk path. These branches or hairs can be thought of as being almost independent, and the lace expansion is used to deal with the intricate dependence structure.

Recently, there is some interest in the branching structure of percolation clusters and lattice trees and animals. In 1993, Aldous [1] conjectured that the scaling limit of lattice trees above 8 dimensions is a process called integrated super-Brownian excursion (ISE). ISE is a process which belongs to the class of super-processes. Super-processes are well-investigated, and have received a lot of attention in the probability community (see e.g., Le Gall [17]).

The relation between super-processes and percolation models was further investigated in [10,11], where it was proved that percolation clusters share features of ISE. In [12], it was proved that large critical oriented percolation clusters in the -dimensional lattice, with , converge after a suitable rescaling, to super-Brownian motion. The same behaviour is expected to be true for the contact process [21].

# 2. Structure

In this workshop, we would like to give probabilists interested in percolation the opportunity to learn about the recent progress in the determination of scaling limits. The aim of the workshop is to bring young researchers together so that they can exchange ideas and initialize cooperations.

Since the progress above described is made in fields that are rather difficult to penetrate for relative outsiders, we think it is important to plan for mini-courses in those directions. Thus, the workshop will incorporate

·                    Mini-courses by a invited researchers;

·                    Talks by the participants about topics related to percolation and scaling limits;

·                    Problem sessions chaired by selected participants, in which new projects for cooperation will be presented.

We will reserve time for discussion and exchange about the presented open problems.

[1]  D. Aldous. Tree-based models for random distribution of mass. J. Stat. Phys., :625–641, (1993).

[2]  D.C. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., :125–148, (1985).

[3]  J.L. Cardy. Conformal invariance and surface critical behaviour. Nucl. Phys. B, : 514–532, (1984).

[4]  J.L. Cardy. The number of incipient spanning clusters in two-dimensional percolation. J. Phys. A, : L105, (1998).

[5]  E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., :69–104, (1998).

[6]  T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., :333–391, (1990).

[7]  T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., :1469–1510, (1990).

[8]  T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys., :101–136, (1992).

[9]  T. Hara and G. Slade. The number and size of branched polymers in high dimensions. J. Stat. Phys., :1009–1038, (1992).

[10]  T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys., :1075–1168, (2000).

[11]  T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys., :1244–1293, (2000).

[12] R. van der Hofstad and G. Slade. Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Ann. Inst. H. Poincaré Probab. Statist.  (2003): 413–485.

[13]  Lawler, G. F., Schramm, O. Werner, W. Values of Brownian intersection exponents, I: Half-plane exponents. Acta Math., : 237–273, (2001).

[14]  Lawler, G. F., Schramm, O., Werner, W. Values of Brownian intersection exponents, II: Plane exponents. Acta Math., :275-308, (2001).

[15]  Lawler, G. F., Schramm, O., Werner, W. Values of Brownian intersection exponents, III: Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., :109-123. (2002).

[16]  Lawler, G. F., Schramm, O., Werner, W. One-arm exponent for critical 2D percolation. Electron. J. Probab., : 13pp (electronic) (2002).

[17]  J.-F.  Le Gall. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics, ETH Zürich, Birkhäuser, 1999.

[18]  K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann. :103-121 (1923).

[19]  B. Nienhuis, E.K. Riedel, M. Schick. Magnetic exponents of the two-dimensional - states Potts model. J. Phys. A, :189-192.

[20]  B. Nienhuis. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Statist. Phys. : 731–761, (1984).

[21] A. Sakai. Mean-field critical behavior for the contact process. J. Statist. Phys.  (2001): 111–143.

[22]  O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., :221–288, (2000).

[23]  S. Smirnov. Critical percolation in the plane. I. Conformal invariance and cardy’s formula. II. Continuum scaling limit. Preprint (2001).

[24]  S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett. , :729-744 (2001).

[25]  G. Slade. The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys., :661-683, (1987).

[26]  G. Slade. Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen., :L417-L420, (1988).

[27]  G. Slade, G. The lace expansion and the upper critical dimension for percolation. Lectures in Applied Mathematics, :53–63, (Mathematics of Random Media, eds. W.E. Kohler and B.S. White, A.M.S., Providence), (1991).

PROGRAMME

Monday-Tuesday-Wednesday-Thursday-Friday

Monday March 29, 2004

 10.00-10.15 Welcome and opening Frank den Hollander 10.15-11.15 Schramm-Loewner Evolutions and scaling limits I Vincent Beffara 11.15-11.30 Coffee/Tea break 11.30-12.30 Schramm-Loewner Evolutions and scaling limits II Vincent Beffara 12.30-15.00 Lunch 15.00-16.00 Inequalities in percolation and related fields Rob van den Berg 16.00-16.15 Coffee/Tea break 16.15-17.00 Gibbsian modifications of polygonal Markov fields in the plane: phase transitions and random dynamics Tomasz Schreiber 17.00-17.45 Mean-field behavior for the survival probability and the percolation point-to-surface connectivity Akira Sakai

Tuesday March 30, 2004

 09.15-10.15 Schramm-Loewner Evolutions and scaling limits III Vincent Beffara 10.15-10.30 Coffee/Tea Break 10.30-11.30 Schramm-Loewner Evolutions and scaling limits IV Vincent Beffara 11.30-12.30 Critical lattice percolation on the strip and cylinder Bernard Nienhuis 12.30-15.00 Lunch 15.00-15.45 Coexistence in two-type first-passage percolation models Règine Marchand 15.45-16.30 Block renormalization for 2d FK-percolation Reda Messik 16.30-17.00 Coffee/Tea Break 17.00-17.45 Some news from the lilypond Stas Volkov

Wednesday March 31, 2004

 09.15-10.15 Schramm-Loewner Evolutions and scaling limits V Vincent Beffara 10.15-10.30 Coffee/Tea Break 10.30-11.30 Schramm-Loewner Evolutions and scaling limits VI Vincent Beffara 11.30-12.30 The incipient infinite cluster for oriented percolation above 4 dimensions Frank den Hollander 12.30-15.00 Lunch 15.00-15.45 "New" invariant Gibbs measures for the Potts model Per Hallberg 15.45-16.30 The Full Scaling Limit of 2D Critical Percolation Federico Camia 16.30-17.00 Coffee/Tea Break 17.00-17.45 Scaling of an exploration process in random planar maps Omer Angel

Thursday April 1, 2004

 09.15-10.15 Motivation and Kesten's construction of the IIC in 2D Antal Járai 10.15-10.30 Coffee/Tea Break 10.30-11.30 Various alternative constructions in 2D Antal Járai 11.30-12.30 Fractal percolation versus classical percolation Ronald Meester 12.30-12.45 Coffee/Tea Break 12.45-13.30 Fractal percolation and set-valued substitutions Peter van der Wal 13.30 Lunch FREE AFTERNOON 17.00 Start excursion to  Bilderdijklaan 10 Eindhoven tel. +31 (0)40 238 10 00 19.00 Dinner in Restaurant "Listers", Kleine Berg 57h, tel. 040-2961370

Friday April 2, 2004

 09.15-10.15 The IIC in high dimensions Antal Járai 10.15-10.30 Coffee/Tea Break 10.30-11.30 Random walk on critical branching process trees and the IIC Antal Járai 11.30-12.30 Exponential laws for rare events in Gibbsian random fields Frank Redig 12.30-12.45 Coffee/Tea Break 12.45-13.30 Dimer models : Scaling limit of edge densities Cedric Boutillier 13.30 Lunch

PRACTICAL INFORMATION

Location

Eurandom is located on the campus of Eindhoven University of Technology, in the 'Laplacegebouw' building' (LG on the map).  The university is located at 10 minutes walking distance from Eindhoven railway station (take the exit north side and walk towards the tall building on the right with the sign TU/e).

Travel

For those arriving by plane, there is a convenient train connection between Amsterdam Schiphol airport and Eindhoven, with only one change at Duivendrecht. This trip will take about an hour and a half. For more detailed information, please consult the NS travel information pages or see EURANDOM web page location.

Hotel

For invitees we will book a room in one of the hotels close to the Eindhoven University of Technology. You will receive a separate confirmation of your hotel reservation from the congress office of the Eindhoven University of Technology after you have filled out the registration form.

Hotel 'De Bengel', Wilhelminaplein 9, Eindhoven, tel. 00 31 40 2440752

Hotel 'Van Neer', Kronehoefstraat 10, Eindhoven, tel. 00 31 40 2433436

Hotel 'Corso', Vestdijk 17, Eindhoven, tel. 00 31 40 2449131

Lunch

On all workshop days lunch will be organised with no costs involved, if you have ordered this on the registration form.

Social event

On Thursday March 1 an excursion to the Van Abbemuseum is planned starting at 5 PM.

Workshop dinner is also planned on this Thursday March 1. Place and time to be announced.

#### Organisers

Nina Gantert (Universität Karlsruhe), Remco van der Hofstad (EURANDOM/Eindhoven University of Technology)

Co-sponsored by  the ESF-RDSES    Programme and the Thomas Stieltjes Institute for Mathematics