**Stochastic
Activity Month**

**February 2012**

**"Scaling
limits in spatial probability"**

SUMMARY |
REGISTRATION |
SPEAKERS |

The study of random spatial
systems is a key field in Statistical Physics. The impact of the underlying
geometry produces rich and fascinating behaviour. Particularly the study of
phase transitions and corresponding scaling limits gives a thorough description
of such behaviour. Obtaining the scaling limit is a
very challenging
task, yet, substantial progress has been made in special cases in
the past years.
In particular, the last decade has seen a number of
breakthroughs that revolutionized our understanding of scaling limits of spatial
systems, both in two as well as in high
dimensions. Most prominently, the identification of
the Schramm-Loewner Evolution enabled a mathematically rigorous
description of scaling limits in two-dimensional spatial models. Moreover, there
are important recent developments in renormalization group techniques, cluster
expansion and the study of high-dimensional systems.

In this Stochastic Activity
Month, we aim at bringing together senior and junior researchers at EURANDOM,
to bundle activities in these fields.

● Friday, February 3, 2012
Opening with a festive **Mark Kac seminar** in Utrecht.
Speakers: Emilio Cirillo, Rob van den Berg.

The following activities will be held at Eurandom, Eindhoven:

● February 7: 11.40 - 12.40h., seminar by Siamak Taati (Utrecht University)

● February 8: 11.40 - 12.40h., seminar by Benedetto Scoppola ((l’Universita’ Roma 2).

● February 8: 14.00 - 17.30h., lecture afternoon "Applications fo Scaling Limits". Speakers: Ronald Meester (VU Amsterdam), Pieter Trapman (Stockholm University).

● February 13: mini-workshop **
Scaling limits of random walks in the quarter plane** organized by Johan van
Leeuwaarden and Kylian Raschel,

● February 14-17: workshop **The expanding art of expansions** on cluster expansion,
lace expansion and renormalization group methods, organized by Roberto
Fernandez, Markus Heydenreich and Remco van der Hofstad.

● February 21: 14.00 - 15.00 h., seminar by Charlene Kalle (University
Leiden): **Local dimensions for the infinite Bernoulli
convolution**

● February 22: ** Lectureday "Scaling limits"**. Speakers are Federico Camia,
Anne Fey, Jesse Goodman Wouter Kager.

● February 23: 14.00 - 15.00 h., seminar by Christoph Temmel (TU Graz):
**Partition schemes tailored to
clusters and new bounds for abstract polymer models**

● February 24: 14.00 - 15.00 h., seminar by Alex Opoku (University Leiden):Copolymer with adsorption: variational characterization of the critical curve

● February 27-March 2: workshop **Two-dimensional statistical mechanics **(in the series
"Young European Probabilists"), organized by Artem Sapozhnikov and Hugo
Duminil-Copin.

Further details on all activities will be updated as they become available.

**VISITORS **

Stefan Adams | Warwick Mathematics Institute |

Erwin Bolthausen | Universität Zürich |

Emilio Cirillo | Università degli Studi di Roma |

Roman Kotecký | Warwick Mathematics Institute; Charles University Prague |

**Alex Opoku
(University Leiden)
**24-02-2012

**Copolymer with adsorption:
variational characterization of the critical curve**

Consider a linear chain (polymer)
that is random concatenation of two different types of blocks (monomers), say
hydrophobic and hydrophilic monomers, located in the vicinity of an impure or
imperfect linear interface separating two immiscible solvents, say oil and
water. The impurities at the interface and the arrangement of the monomer types
along the chain are modeled by i.i.d. sequences of random variables. The
configurations of the polymer are directed paths that can make i.i.d. excursions
of finite length above and below the interface.

The model of interest is driven by two types of interactions, namely; (1)
monomer-solvent interaction and (2) monomer-interface interaction. This model
undergoes a localization (polymer stays close to the interface)-delocalication
(polymer stays away from the interface) phase transition in the parameter space
of the model with a critical curve separating the two phases. Not much is known
about the precise shape of this critical curve. We report on some recent
progress in this regard.

**Christoph
Temmel**

23-02-2012

**Partition schemes tailored to
clusters and new bounds for abstract polymer models**

The classic lower bound for the convergence of the cluster expansion of an abstract polymer system around zero fugacity is the Dobrushin-condition. There have been two separate improvements of this condition. Scott & Sokal pointed the way to a reduction of the degree in the condition by one, while keeping the simple inductive proof. On the other hand Fernàndez & Procacci applied tree-operator techniques to the cluster expansion to use more information about the local neighbourhood structure of the system. These two improvements are not directly comparable. We apply these tree-operator techniques to derive a degree-reduced condition incorporating the local neighbourhood structure of the system. The key ingredient is a partition of the connected subgraph complex of a cluster adapted to the structure of the cluster. We then sketch ongoing work to transfer this to the continuous hard sphere model and create a condition simultaneously improving on all known conditions.

**Charlene Kalle**

**Local dimensions for the
infinite Bernoulli convolution**

The infinite Bernoulli convolution is a probability measure obtained by convolving infinitely many Bernoulli measures. We will use a specific dynamical system, called the `random beta-transformation', to study the local behaviour of this mysterious measure.

**Benedetto
Scoppola (**l’Universita’
Roma 2)

08-02-2012

**Phase Transitions for the
Cavity Approach to the Clique Problem on Random Graphs**

We will discuss a MCMC algorithm to find large cliques on random Erdos graphs. The algorithm is inspired by the idea of cavity fields, introduced in the context of disordered systems, and is a conservative probabilistic cellular automata. The dynamics of this MCMC exhibits a quite interesting phase diagram, that is possible to study rigorously, proving the existence of two phase transitions. This phase diagram has important consequences on the performances of the algorithm, and they will be briefly discussed. This is a joint work with Alexander Gaudilliere (CNRS Marseille) Elisabetta Scoppola (Universita' Roma tre) and Massimiliano Viale (Universita' La Sapienza di Roma).

**Siamak Taati
(Utrecht University)**

07-02-2012

**Discrete approximations of
continuum gas models (or vice versa)**

I will talk about our recent work in progress on the connection between the hardcore gas models in the continuum and on the lattice.

**Christoph
Temmel**

**Partition schemes tailored to
clusters and new bounds for abstract polymer models**

The classic lower bound for the convergence of the cluster expansion of an abstract polymer system around zero fugacity is the Dobrushin-condition. There have been two separate improvements of this condition. Scott & Sokal pointed the way to a reduction of the degree in the condition by one, while keeping the simple inductive proof. On the other hand Fernàndez & Procacci applied tree-operator techniques to the cluster expansion to use more information about the local neighbourhood structure of the system. These two improvements are not directly comparable. We apply these tree-operator techniques to derive a degree-reduced condition incorporating the local neighbourhood structure of the system. The key ingredient is a partition of the connected subgraph complex of a cluster adapted to the structure of the cluster. We then sketch ongoing work to transfer this to the continuous hard sphere model and create a condition simultaneously improving on all known conditions.