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Workshop
on
THE EXPANDING ART OF EXPANSIONS
February 14 - 17, 2012
SUMMARY | REGISTRATION | SPEAKERS |
ABSTRACTS |
PARTICIPANTS |
Expansion techniques are a strong technical tool in the study of random spatial systems.
Though some of the techniques are nowadays considered classical, there are fascinating recent developments shedding new light on these "classical" tools. During the workshop we aim to highlight these fascinating new insights and particularly the links between them.
This workshop brings together leading experts for the various expansion flavors, as well as talented junior researchers.
The workshop is part of the Stochastic Activity Month (SAM) "Scaling limits in Spatial Probability", taking place at Eurandom through February 2012. Workshop participants are encouraged to extend their visit and benefit from the various SAM activities.
More information about the Stochastic Activity Month can be found on this website:
http://www.eurandom.nl/events/workshops/2012/SAM2/indexSAM2.htmFields covered by the workshop:
Cluster Expansion - Lace Expansion - Renormalization Group Methods - Gradient Models
Confirmed speakers:
Stefan Adams | Warwick Mathematics Institute |
Luca Avena | Universität Zürich |
Erwin Bolthausen | Universität Zürich |
Emilio Cirillo | Università degli Studi di Roma |
Codina Cotar | Fields Institute Toronto |
Jean-Dominique Deuschel | Technische Universitat Berlin |
Pierluigi Falco | California State University |
Robert Fitzner | TU Eindhoven |
Markus Heydenreich | CWI - Leiden University |
Frank den Hollander | University of Leiden |
Mark Holmes | University of Auckland |
Tim Hulshof | TU Eindhoven |
Roman Kotecký | Warwick Mathematics Institute; Charles University Prague |
Gady Kozma | Weizman Institut Rehovot |
Takashi Kumagai | Kyoto University |
Daisuke Shiraishi | Kyoto University |
Roberto Fernandez (Universiteit Utrecht)
Markus Heydenreich (Universiteit Leiden and CWI Amsterdam)
Remco van der Hofstad (Technische Universiteit Eindhoven and Eurandom)
Speakers and participants of this workshop are kindly invited to extend their stay at Eurandom by a few (or more) days in the context of the Stochastic Activity Month (link to SAM website). Please contact one of the organizers when considering this possibility.
TUESDAY Feb 14
09.30 - 10.10 | Registration - Coffee/tea | |
10.10 - 10.15 | Opening by Remco van der Hofstad | |
10.15 - 11.15 | Frank den Hollander | Renormalisation of hierarchically interacting Cannings processes |
11.15 - 12.15 | Emilio Cirillo | Graded Cluster Expansion |
12.15 - 14.00 | Lunch | |
14.00 - 15.00 | Takashi Kumagai | Quenched invariance principle for random walks and random divergence forms in random media on cones |
15.00 - 15.30 | Coffee/tea break | |
15.30 - 16.30 | Daisuke Shiraishi | Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions |
16.30 - 17.30 | Tim Hulshof | Random walk on the incipient infinite cluster |
WEDNESDAY Feb 15
09.00 - 10.00 | Markus Heydenreich | A new lace expansion for high-dimensional incipient infinite cluster |
10.00 - 10.30 | Coffee/tea break | |
10.30 - 11.30 | Robert Fitzner | Non-backtracking lace expansion for nearest-neighbor models |
11.30 - 12.30 | Mark Holmes | Convergence of high dimensional models to super-Brownian motion |
12.30 - 13.30 | Lunch | |
13.30 - | Free afternoon | |
18.30 - | Conference dinner |
THURSDAY Feb 16
09.00 - 10.00 | Pierluigi Falco | Statistical Mechanics of the Two-Dimensional Coulomb Gas |
10.00 - 10.30 | Coffee/tea break | |
10.30 - 11.30 | Stefan Adams | Random field of gradients - a RG approach |
11.30 - 12.30 | Roman Kotecky | From gradient Gibbs measures to nonlinear elasticity |
12.30 - 13.30 | Lunch | |
13.30 - | Free time | |
15.30 - 16.30 | Jean-Dominique Deuschel | Hydrodynamic limit for the Ginsburg Landau model with non convex potential |
16.30 - 17.30 | Codina Cotar | Uniqueness of random gradient states |
FRIDAY Feb 17
09.00 - 10.00 | Gady Kozma | Random walks on discrete groups |
10.00 - 10.30 | Coffee/tea break | |
10.30 - 11.30 | Luca Avena | Local central limit theorem estimates for Weakly Self-Avoiding Walks in continuous space |
11.30 - 12.30 | Erwin Bolthausen | Weakly self-avoiding asymmetric random walks |
12.30 - | Closing |
ABSTRACTS
Stefan Adams
Random field of gradients - a RG approach
Random fields of gradients are a class of model systems
arising in the studies of random interfaces, random geometry, field theory, and
elasticity theory. These random objects pose challenging problems for
probabilists as even an a priori distribution involves strong correlations.
Gradient fields are likely to be an universal class of models combining
probability, analysis and physics in the study of critical phenomena. They
emerge in the following three areas, effective models for random interfaces,
Gaussian Free Fields (scaling limits), and mathematical models for the
Cauchy-Born rule of materials, i.e., a microscopic approach to nonlinear
elasticity. The latter class of models requires that interaction energies are
non-convex functions of the gradients. Open problems include unicity of Gibbs
measures and strict convexity of the free energy. We present in the talk a first
break through for the free energy at low temperatures using Gaussian measures
and rigorous renormalisation group techniques. The key ingredient is a finite
range decomposition for parameter dependent families of Gaussian measures.
(joint work with R. Kotechky and S. Mueller)
Luca Avena
Local central limit theorem estimates for Weakly Self-Avoiding Walks in continuous space
We discuss a general method to obtain a Local Central
Limit Theorem (LCLT) for distributions defined by certain renewal type
equations.
This method was first introduced in 2001 by E.Bolthausen and C.Ritzmann to
obtain a LCLT for the usual Weakly Self-Avoiding Walk (WSAW) above four
dimensions. The key idea is to show that the fixed point of a suitable
contractive operator remains asymptotically close to a normal distribution. In
particular, such a method does not make use of Laplace or Fourier transforms.
We extend the mentioned LCLT in the case of WSAW on R^d instead of the usual
Z^d. Working in the continuous instead of the discrete space has the advantage
that several bounds and estimates for convolutions become easier, considerably
simplifying the application of the general method.
(joint work with E. Bolthausen)
Erwin Bolthausen
We consider weakly self-avoiding random walks on the
d-dimensional lattice without assuming that the one-step distribution of the
random walk is symmetric. We apply the classical lace expansion, and a variant
of the contraction method developed by Bolthausen and Christine Ritzmann. Due to
some technical difficulties, the method applies currently only to d=9 and higher.
(joint work with Felix Rubin and Christine Ritzmann)
Emilio Cirillo
Graded Cluster Expansion
A Cluster Expansion approch to disordered systems is discussed. The talk will focus on an iterative expansion method aimed to compute the partition function of a Statistical Mechanics model with large interactions in arbitrarly large but well separated regions. It will shown how this regions have to been classified and iteratively treated in order to get a convergent Cluster Expansion.
Codin Cotar
Uniqueness of random gradient states
We consider two versions of random gradient models. In model A)
the interface feels a bulk term of random fields while in model B) the disorder
enters though the potential acting on the gradients itself. It is well known
that without disorder there are no Gibbs measures in infinite volume in
dimension d = 2, while there are gradient Gibbs measures describing an
infinite-volume distribution for the increments of the field, as was shown by
Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in
model A) prohibits the existence of such gradient Gibbs measures for general
interaction potentials in d = 2. Cotar and Kuelske proved the existence of
shift-covariant gradient Gibbs measures for model A) when d\ge 3 and the
expectation with respect to the disorder is zero, and for model B) when d\ge 2.
In the current work, we prove uniqueness of shift-covariance gradient Gibbs
measures with given tilt under the above assumptions.
(joint work with Christof Külske)
Jean-Dominique Deuschel
Hydrodynamic limit for the Ginsburg Landau model with non convex potential
We consider the dynamic of an effective interface model
with gradient interaction and derive the corresponding hydrodynamical limit,
extending the result of [FS] to a certain class of non convex interaction
potential. This includes the high temperature regime, where uniqueness of
ergodic gradient Gibbs maesures has been proved recently in [CD]. Our proof is
based on the identification of the translation invariant stationary measures as
gradient Gibbs measures.
(joint work with T Nishikawa and Y Vignaud)
[CD] C.Cotar and J.-D.Deuschel, Decay of covariances, uniqueness of ergodic component and scaling limit for a class of $\nabla\phi$ systems with non-convex potential,Ann. Inst. H. Poincar\'e Probab. Statist.(2012).
[FS] T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau $\nabla\phi$ interface model,Commun.Math.Phys.(1997)
Pierluigi Falco
Statistical Mechanics of the Two-Dimensional Coulomb Gas
The Coulomb gas is the prototype of probabilistic model displaying a special kind of phase transition, the 'Kosterlitz-Thouless' one. In this presentation, I will review conjectures, results and works in progress on this model.
Robert Fitzner
Non-backtracking lace expansion for nearest-neighbor models
In this talk we introduce the non-backtracking lace expansion (NBLE), that we developed to enhance known results in the nearest-neighbor setting on the Zd-lattice. We first review results for self-avoiding walk and percolation, obtained using the classical lace expansion. Then we present the idea of the non-backtracking lace expansion and sketch the analysis of the lace expansion.
Markus Heydenreich
A new lace expansion for high-dimensional incipient infinite cluster
Incipient infinite cluster (IIC) is a critical percolation cluster conditioned on infinite size. It is a natural example of a spatial object showing fractal properties, self-similarity, and a non-degenerate scaling limit. Kesten (1986) proved its existence on the two-dimensional lattice. This talk concentrates on incipient infinite clusters in high-dimension, whose existence was proven by Van der Hofstad and Jarai (2004). I review the lace expansion for percolation by Hara and Slade (1990), and explain a new lace expansion approach for the backbone of the incipient infinite cluster. This new approach allows the identification of the scaling limit of the IIC backbone. The talk is based on joint work with R. van der Hofstad, T. Hulshof and G. Miermont.
Frank den Hollander (Leiden University)
Renormalisation of hierarchically interacting Cannings
processes
(joint work with Andreas Greven, Sandra Kliem and Anton Klimovsky)
In order to analyse universal patterns in the large space-time behaviour of interacting multi-type stochastic populations, a key approach has been to carry out a renormalisation analysis. This has provided considerable insight into the structure of systems of interacting diffusions.
In this talk we describe a system of hierarchically
interacting Cannings processes. The latter are jump process that arise as the
continuum limit of a population model in which the offspring of a single
individual can be a positive fraction of the total population. The interaction
between the individuals comes from migration and resampling on all hierarchical
space-time scales simultaneously. Individuals live in colonies labelled by the
hierarchical group $\Omega_N$ of order $N$, and are subject to migration and
resampling in $k$-blocks for all $k\in\N_0$ based on a sequence of migration
rates $\uc=(c_k)_{k\in\N_0}$ and a sequence of offspring measures $\uL= (\Lambda_k)_{k\in\N_0}$.
For this system we carry out a full renormalisation analysis in the hierarchical
mean-field limit $N \to \infty$.
Our main result is that, in the limit as $N\to\infty$, on each hierarchical
scale $k\in\N_0$ the $k$-block averages of the system converge to a random
process that is a superposition of a single-component Cannings process and a
single-component Fleming-Viot process, the latter with a drift $c_k$ and with a
volatility constant $d_k$ that turns out to be a function of $c_l$ and $\Lambda_l$
for all $0 \leq l <k$. It is through the volatility that the renormalisation
manifests itself.
We investigate how the volatility scales as $k\to\infty$, which requires an
analysis of iterations of certain M\"obius-transformations.
We discuss the implications of this scaling for the behaviour on large
space-time scales, comparing the outcome with what is known from the
renormalisation analysis of hierarchically interacting Fleming-Viot diffusions,
pointing out several new features.
Mark Holmes (Auckland University)
Convergence of high dimensional models to super-Brownian
motion
(joint work with van der Hofstad and Perkins)
The scaling limit of critical branching random walk is a measure-valued process called super-Brownian motion (SBM). Various self-repelling (critical) models such as the contact process, oriented percolation, and lattice trees are conjectured to converge to SBM above their respective critical dimensions. In each case the lace expansion has been used to prove convergence of the finite-dimension distributions. We now believe that the lace expansion can be used to prove tightness as well.
Tim Hulshof
Random walk on the incipient infinite cluster
The incipient infinite cluster (IIC) is a critical
percolation conditioned to have an infinite cluster at the origin. It is
believed that an approach to analyzing the structure of critical percolation
clusters can be made by studying the behavior of simple random walk on the
incipient infinite cluster. I will discuss properties of random walk on
high-dimensional IICs, as well as properties of the random walk on the `backbone’
of the IIC. In the final part of the lecture I will discuss the specific case of
random walk on the IIC of long-range percolation, where the edge-probabilities
decay as a power-law of the length of the edge. In this setting the behavior of
random walk changes significantly.
(joint work with Markus Heydenreich and Remco v.d. Hofstad)
Roman Kotecky
From gradient Gibbs measures to nonlinear elasticity
The links between macroscopic
elasticity with variational principles articulated in terms of nonlinear elastic
free
energy and equilibrium gradient models are discussed. The non-convexity of the
free energy will be discussed and results concerning large
deviation principle and asymptotic behaviour in terms of gradient Young-Gibbs
measures will be formulated.
(joint work with S. Luckhaus)
Gady Kozma
Random walks on discrete groups
We will survey random walks on discrete groups with an emphesis on groups with polynomial growth. No new results will be presented. No familiarity with the topic will be assumed.
Takashi Kumagai
Quenched invariance principle for random walks and
random divergence forms in random media on cones
(joint work with Z.Q. Chen (Seattle) and D.A. Croydon (Warwick))
We will consider the following two models and establish
quenched invariance principles;
1. Simple random walks on the infinite clusters for super-critical percolations
on half and quarter planes
in d-dimensional Euclidean spaces.
2. Uniform elliptic divergence forms with random stationary coefficients on
cones in Euclidean spaces.
Note that because of the lack of translation invariance, we cannot apply the
method of the 'corrector'.
Instead, we make full use of the heat kernel estimates and Dirichlet form
techniques to resolve the problem.
Daisuke Shiraishi
Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions
We consider two random walks conditioned “never to intersect” in
Z^{2} . We
show that each of them has inﬁnitely many global cut times with
probability one. In fact, we prove that the number of global cut times up to
n grows like
n ^{3/8} . Next we
consider the union of their trajectories
to be a random subgraph of Z^{2}
and show the subdiffusivity of the simple random walk on this graph.
NAME | FIRSTNAME | AFFILIATION |
Adams | Stefan | University of Warwick |
Avena | Luca | Universität Zürich |
Bethuelsen | Stein | Utrecht University |
Bolthausen | Erwin | Universität Zürich |
Cirillo | Emilio | Universita` d Roma |
Cotar | Codina | The Fields Institute |
den Hollander | Frank | University of Leiden |
Deuschel | Jean-Dominique | TU Berlin |
Falco | Pierluigi | California State University Northridge |
Fernandez | Roberto | Utrecht University |
Fitzner | Robert | TU Eindhoven |
Heydenreich | Markus | Universiteit Leiden + CWI Amsterdam |
Holmes | Mark | University of Auckland |
Hulshof | Tim | TU Eindhoven - Eurandom |
Kager | Wouter | VU Amsterdam |
Klimovskiy | Anton | Eurandom |
Kotecky | Roman | University of Warwick |
Kozma | Gady | Weizmann Institute |
Kraaij | Richard | VU Amsterdam |
Kumagai | Takashi | Kyoto University |
Nardi | Francesca | TU Eindhoven |
Ruszel | Wioletta | TU Eindhoven |
Shiraishi | Daisuke | Kyoto University |
Spitoni | Cristian | LUMC Leiden |
Tate | Stephen | University of Warwick |
Van der Hofstad | Remco | TU Eindhoven - Eurandom |
Conference Location
The workshop location is Eurandom, Den Dolech 2, 5612 AZ
Eindhoven, Laplace Building, 1st floor, LG 1.105.
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).
For all information on how to come to Eindhoven, please check http://www.eurandom.tue.nl/contact.htm
CONTACT
For more information please contact
Mrs. Patty Koorn,
Workshop officer of Eurandom
Last updated
24-10-12,
by
PK