YEP 2004 (Young European Probabilists)
workshops
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
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.
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(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).
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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
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[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.
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[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.
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[27] G. Slade, G. The lace
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