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THE EXPANDING ART OF EXPANSIONS
 

February 14 - 17, 2012

SUMMARY

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:

Fields covered by the workshop:

Cluster Expansion - Lace Expansion - Renormalization Group Methods - Gradient Models

SPEAKERS

Confirmed speakers:

ORGANIZERS

Roberto Fernandez (Universiteit Utrecht)

Markus Heydenreich (Universiteit Leiden and CWI Amsterdam)

Remco van der Hofstad (Technische Universiteit Eindhoven and Eurandom)

REGISTRATION

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.

PROGRAMME

TUESDAY Feb 14

WEDNESDAY Feb 15

THURSDAY Feb 16

FRIDAY Feb 17

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)

PRESENTATION

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)

PRESENTATION

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 Z² . 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 Z² and show the subdiffusivity of the simple random walk on this graph.

PARTICIPANTS

PRACTICAL INFORMATION

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