YEP (Young European Probabilists)
Workshop on "Conformal invariance, scaling limits and percolation"
EURANDOM, Eindhoven, The Netherlands
March 29 to April 2, 2004
Programme &Abstracts Speakers & Participants Registration  Practical Information
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 twodimensional 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
twodimensional 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 SchrammLoewner
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 spacefilling.
Originally, Schramm [22] used SLE to prove that the scaling limits of
looperased random walk is SLE_{2} 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 wellknown 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 twodimensional 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
onearm exponent [16], and the critical exponent for subcritical cluster sizes
[24].
The lace expansion for highdimensional 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 endpoint of
weakly selfavoiding walk behaves diffusively. Weakly selfavoiding walk can be
seen as simple random walk of a fixed length that is penalized by a factor _{} for every
selfintersection. The lace expansion is a combinatorial resummation identity
that relates (weakly) selfavoiding walks of length _{} to selfavoiding
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) selfavoiding 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 selfavoiding 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 selfavoiding 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 selfavoiding 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 inclusionexclusion, due to the fact
that the interaction for percolation is more deeply hidden.
Initially, Hara and Slade only investigated socalled twopoint
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 superBrownian excursion (ISE). ISE is a process
which belongs to the class of superprocesses. Superprocesses are
wellinvestigated, and have received a lot of attention in the probability
community (see e.g., Le Gall [17]).
The relation between superprocesses 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 superBrownian
motion. The same behaviour is expected to be true for the contact process [21].
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 minicourses in those directions. Thus, the workshop will incorporate
·
Minicourses 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. Treebased
models for random distribution of mass. J. Stat. Phys., :625–641,
(1993).
[2] D.C. Brydges and
T. Spencer. Selfavoiding 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 twodimensional 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. Meanfield 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. Selfavoiding 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
highdimensional 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
highdimensional percolation. II. Integrated superBrownian excursion. J.
Math. Phys., :1244–1293, (2000).
[12] R. van der Hofstad and G. Slade. Convergence of critical oriented
percolation to superBrownian 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: Halfplane exponents. Acta
Math., : 237–273, (2001).
[14] Lawler, G. F., Schramm, O.,
Werner, W. Values of Brownian intersection exponents, II: Plane exponents. Acta
Math., :275308, (2001).
[15] Lawler, G. F., Schramm, O.,
Werner, W. Values of Brownian intersection exponents, III: Twosided exponents.
Ann. Inst. H. Poincaré Probab. Statist., :109123. (2002).
[16] Lawler, G. F., Schramm, O.,
Werner, W. Onearm 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. :103121 (1923).
[19] B. Nienhuis, E.K. Riedel,
M. Schick. Magnetic exponents of the twodimensional _{} states Potts model. J. Phys. A, :189192.
[20] B. Nienhuis. Critical
behavior of twodimensional spin models and charge asymmetry in the Coulomb
gas. J. Statist. Phys. : 731–761, (1984).
[21] A. Sakai. Meanfield critical behavior for the contact process. J.
Statist. Phys. (2001): 111–143.
[22] O. Schramm. Scaling
limits of looperased 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 twodimensional percolation. Math.
Res. Lett. , :729744 (2001).
[25] G. Slade. The
diffusion of selfavoiding random walk in high dimensions. Commun. Math.
Phys., :661683, (1987).
[26] G. Slade. Convergence
of selfavoiding random walk to Brownian motion in high dimensions. J. Phys.
A: Math. Gen., :L417L420, (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).
MondayTuesdayWednesdayThursdayFriday
Monday March 29, 2004
10.0010.15  Welcome and opening  Frank den Hollander 
10.1511.15  SchrammLoewner Evolutions and scaling limits I  Vincent Beffara 
11.1511.30  Coffee/Tea break  
11.3012.30  SchrammLoewner Evolutions and scaling limits II  Vincent Beffara 
12.3015.00  Lunch  
15.0016.00  Inequalities in percolation and related fields  Rob van den Berg 
16.0016.15  Coffee/Tea break  
16.1517.00  Gibbsian modifications of polygonal Markov fields in the plane: phase transitions and random dynamics  Tomasz Schreiber 
17.0017.45  Meanfield behavior for the survival probability and the percolation pointtosurface connectivity  Akira Sakai 
Tuesday March 30, 2004
09.1510.15  SchrammLoewner Evolutions and scaling limits III  Vincent Beffara 
10.1510.30  Coffee/Tea Break  
10.3011.30  SchrammLoewner Evolutions and scaling limits IV  Vincent Beffara 
11.3012.30  Critical lattice percolation on the strip and cylinder  Bernard Nienhuis 
12.3015.00  Lunch  
15.0015.45  Coexistence in twotype firstpassage percolation models  Règine Marchand 
15.4516.30  Reda Messik  
16.3017.00  Coffee/Tea Break  
17.0017.45  Some news from the lilypond  Stas Volkov 
Wednesday March 31, 2004
09.1510.15  SchrammLoewner Evolutions and scaling limits V  Vincent Beffara 
10.1510.30  Coffee/Tea Break  
10.3011.30  SchrammLoewner Evolutions and scaling limits VI  Vincent Beffara 
11.3012.30  The incipient infinite cluster for oriented percolation above 4 dimensions  Frank den Hollander 
12.3015.00  Lunch  
15.0015.45  "New" invariant Gibbs measures for the Potts model  Per Hallberg 
15.4516.30  The Full Scaling Limit of 2D Critical Percolation  Federico Camia 
16.3017.00  Coffee/Tea Break  
17.0017.45  Scaling of an exploration process in random planar maps  Omer Angel 
Thursday April 1, 2004
09.1510.15  Motivation and Kesten's construction of the IIC in 2D  Antal Járai 
10.1510.30  Coffee/Tea Break  
10.3011.30  Various alternative constructions in 2D  Antal Járai 
11.3012.30  Fractal percolation versus classical percolation  Ronald Meester 
12.3012.45  Coffee/Tea Break  
12.4513.30  Fractal percolation and setvalued substitutions  Peter van der Wal 
13.30  Lunch  
FREE AFTERNOON  
17.00

Start excursion to
Bilderdijklaan 10 Eindhoven 

19.00 
Dinner in Restaurant "Listers", Kleine Berg 57h, tel. 0402961370 
Friday April 2, 2004
09.1510.15  The IIC in high dimensions  Antal Járai 
10.1510.30  Coffee/Tea Break  
10.3011.30  Random walk on critical branching process trees and the IIC  Antal Járai 
11.3012.30  Exponential laws for rare events in Gibbsian random fields  Frank Redig 
12.3012.45  Coffee/Tea Break  
12.4513.30  Dimer models : Scaling limit of edge densities  Cedric Boutillier 
13.30  Lunch 
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.
For more information please contact Mrs. Lucienne Coolen, the workshop coordinator of EURANDOM, at coolen@eurandom.tue.nl
Nina Gantert (Universität Karlsruhe), Remco van der Hofstad (EURANDOM/Eindhoven University of Technology)
Cosponsored by the ESFRDSES Programme and the Thomas Stieltjes Institute for Mathematics