RSS

We will study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (a>2).

'Marcelo R. Hilário (IMPA - Rio de Janeiro, Brazil), October 9, 2009
Joint work with: O. Luidor, C. M. Newman, L. T. Rolla, S. Sheffield and V. Sidoravicius

Fixation for Distributed Clustering Processes.

We study a discrete-time resource flow in the d-dimensional Euclidean Lattice. In time $t=0$
a random amount of a resource is given to each vertex. Then at each step, wealthier vertices attract the
resources of their less rich neighbors. We prove that, for any translation-ivariant probability distribution
of initial resource quantities, the flow at each vertex stops at finitely many steps. This answers a
(generalized version) of a question posed by Van den Berg and Meester in 1991.
Our proof use the mass-transport principle.

Random walks in random environment were intensively studied in dimension 1 but are less well understood in higher dimensions. A more reachable problem concerns random walks in random environment on Galton-Watson trees. The talk will focus on the transient case, and especially on the rate escape of the walk. I will present the different regimes, and will discuss large deviations on the asymptotic speed.

The problem of self-avoiding walks has been an interesting and fruitful research field in the last decades. One possibility to understand this problem is to analyze walks with a given memory, that are random walks that are restricted not to self-intersect within a prescribed number of steps. I want to present an approach to evaluate walks with a given memory and by investigating in detail memory-2 walks, that is, random walks without direct reversals. In particular, I shall describe how to obtain a CLT for the end-to-end displacement from the eigenvalue expansion of the transfer matrix. Joint work with Remco van der Hofstad.

Bernardo Nunez Borges de Lima (Universidade Federal de Minas Gerais, Brazil), April 24, 2009
Joint work with Remy Sanchis and Roger Silva

Consider an independent site percolation model on Z^d, where all long range connections in the axes are allowed. We show that given any p (the percolation density), there exists an integer K(p) such that all binary sequences are seen simultaneously, almost surely, even if all connections whose length is bigger than K(p) are supressed. We also show how K(p) scales with p when p goes to zero. Related results are obtained concerning the question of whether or not almost all word are seen.

We consider universality problems on sets of interlaced particles. We construct a number of measures on interlaced particle configurations, both on a line and on a circle. We prove a determinantal structure, and compute the associated space-time correlation kernels. Scaling appropriately, we consider the asymptotic behavior of such systems, as the size of the configurations increase. In the bulk of the configurations, we will see that the correlation kernels asymptotically approach the sine kernel.

In many situations of interest, the potential theoretic approach to metastability allows to derive sharp estimates for quantities characterizing the metastable behavior of a given system. In this framework, the average metastable times can be expressed through the capacity of corresponding metastable sets, and capacities can be estimated with the application of two different variational principles, providing upper and lower bounds. After recalling these basic concepts and techniques, I will describe a new method to couple the dynamics inside potentials wells. Under some general hypothesis, I will show that this yields sharp estimates on metastable times, pointwise on any metastable set. Our key example will the random field Curie-Weiss model.

In many real-world networks, such as the Internet and social networks, power-law degree sequences have been observed. This means that, when the graph is large, the proportion of vertices with degree $k$ is asymptotically proportional to $k^{-\tau}$, for some $\tau \geq 1$. These networks are often small worlds, which means that distances in these networks are small. We will study two random graph models, the configuration model and the preferential attachment model, which will have power-law degree sequences when the number of vertices tends to infinity. An overview is given of known results about distances in these graph models. Also some new results will be presented, among which a $\log \log$ lower bound on the diameter of preferential attachment graphs with $\tau > 2$.

We begin with a simple random walk on (Z/NZ)^d, the d-dimensional integer torus with side length N for d >= 3, stopped at a time of order N^d. For large N, the integer torus locally looks like Z^d. What about the random subset of sites visited by the random walk? In the first part of the talk, we will see that this set locally looks like a random interlacement, a model that has recently been introduced by Sznitman. In the second part of the talk, we will see similar convergence results for several discrete cylinders of the form G_N x Z, where G_N is a sequence of finite connected weighted graphs of diverging size.

First we shall discuss uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. Next, we investigate the long-term behaviour of a particular system of SDEs for $d \geq 2$ types, involving catalytic branching and mutation between types. It can be shown that the overall sum of masses converges to zero but does not hit zero in finite time a.s. Finally, results on the relative behaviour of types are given to obtain some insight on how the process approaches zero.

Shankar Bhamidi (University of California, Berkeley, USA), December 5, 9 and 16, 2008
Joint work with David Aldous, Steve Evans and Arnab Sen.

The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) to the local neighborhood of limiting inﬁnite ob jects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations.

Here we shall give a wide range of examples of the above methodology. In particular

1. We shall show how the above methodology can be used to tackle problem of ﬂows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models how the above methodology allows us to prove the convergence of the empirical distribution of edge ﬂows exhibiting how macroscopic order emerges from microscopic rules. 2. We shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees from a wide variety of attachment schemes etc), how the above methodology shows the convergence of the spectral distribution of the adjacency matrix to a limiting non random distribution function. 3. Time permitting we shall also show how how one can deduce convergence of the maximal matching for various families of random trees and what this means about the spectral mass at 0.

Martin Hutzenthaler (Research assistant in the Dutch-German DFG/NWO research group "Mathematics of Random Spatial Models from Physics and Biology"), November 24, 2008

each other in each colony. Of special interest is a model with N colonies and uniform migration, that is, the target colony is chosen randomly in each migration step. Initially N-1 colonies are empty. As N tends to infinity, this model converges to the Virgin Island Model in which every emigrant populates an empty colony. Furthermore it turns out that the Virgin Island Model is optimal for the total population size among all branching-migration models with identical parameters for branching and competition. This supports the intuition that if there is competition among individuals and resources are everywhere the same, then the best strategy for survival is to move to unpopulated colonies. In addition there is a fairly explicit criterion for survival versus extinction of the total mass of the Virgin Island process.

As part of the research program of the Honours Programme 2007-2008, we, as a group of 4 students, studied complex networks. Complex networks are all around and understanding their structure can help increase efficiency of real networks. We studied an adapted version of the preferential attachment model by Barabási-Albert in which the clustering can be varied and we showed that this adapted model still has a power-law degree sequence with power-law exponent that we determined. Clustering in the old model was significantly lower than in real networks but the adapted model shows promising results of higher clustering. Using data from Rijkswaterstaat we further investigate whether the Dutch road network has some characteristics of a complex network.

We study the limiting behaviour of the large sums of strongly correlated exponentials as the number of their summands and the effective dimension of the correlation structure simultaneously tend to infinity. We consider two types of such sums which are generated by two a priori very different Gaussian correlation structures. The first type is a sum of hierarchically correlated random variables which is based on the partition function of Derrida's generalised random energy model with external field. The second type is an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables which is based on the partition function of the Sherrington-Kirkpatrick model with multidimensional spins. We consider the asymptotic behaviour (the thermodynamic limit) of these two sums on a logarithmic scale (i.e., at the level of free energy) and also at a more refined level of their fluctuations (i.e., at the level of weak limiting laws). Interestingly for the Sherrington-Kirkpatrick model with multidimensional spins, we find traces of a hierarchical organisation in the thermodynamic limit. This supports the conjectured in theoretical physics universal behaviour of the sums of such sort.

Given a graph , a proper vertex colouring of is -frugal if no colour appears more than times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed studied proper -frugal colourings and Yuster studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. In particular, we consider vertex colourings that are not necessarily proper, and in this case, we find qualitative connections with colourings that are -improper — colourings in which the colour classes induce subgraphs of maximum degree at most — for choices of near
to .

We study invasion percolation in two dimensions. We compare properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. We use tools from near-critical percolation in our analysis. This is a joint work with Michael Damron and Balint Vagvolgyi.

Suitably conditioned Brownian motion can be used to study Hamiltonians of quantum particles under holonomic constraints. We give some examples and discuss some of the related problems.

Current models for complex networks mainly aim to reproduce a number of graph properties observed in real-world networks. On the other hand, experimental and heuristic treatments of real-life networks operate under the tacit assumption that the network is a visible manifestation of an underlying hidden reality. For example, it is commonly assumed that communities in a social network can be recognized as densely linked subgraphs, or that Web pages with many common neighbours contain related topics. Such assumptions apply that there is an a priori "community structure" or "relatedness measure" of the nodes, which is reflected bythe link structure of the graph. A common method to represent "relatedness" of objects is by an embedding in a metric space, so that related objects are placed close together, and communities are represented by clusters of points. In this talk, I will discuss graph models where the nodes correspond to points in space, and the stochastic process forming the graph is influenced by the position of the nodes in the space.

The work presented was done in collaboration with William Aiello, Anthony Bonato, Colin Cooper, and Pawel Pralat.

Milan Bradonjic, (UCLA and LANL, USA)
Joint work with Aric Hagberg and Allon G. Percus

We analyze the structure of random graphs generated by the geographic threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, diameter and clustering coefficient are related to the weight distribution and threshold values.

Frank Redig (Leiden University ), May 16, 2008
Based on joint work with J.R. Chazottes and P. Collet (Paris).

We show how, starting from concentration inequalities, one can quite straighforwardly obtain bounds for the speed of relaxation to equilibrium (in L^p) for interacting particle systems. We will also discuss the weak Poincare inequality and how it can be applied in the context of disordered systems and systems at low temperature.

In many real-life networks, power-laws have been observed for the degree sequence, that is, the fraction of vertices with degree $k$ falls of as an inverse power of $k$. Furthermore, many networks are highly clustered, meaning that there is a large number of triangles and other short cycles. I will describe a random graph model, based on so-called random intersection graphs, where both the degree distribution and the clustering can be controlled. If time permits I will also describe some results on how the outcome of an epidemic on such a graph is affected by the clustering.

Balint Virag (University of Toronto), April 25, 2008
This is joint work with B. Valko

Improving a prediction of Wigner, Dyson (1962) gave a formula for the asymptotic probability of large gaps between beta ensemble eigenvalues.

We prove a slightly modified version of Dyson's predictions. The proof relies on the Brownian carousel, a continuum limit of the corresponding random matrices.

Suitably conditioned Brownian motion can be used to study Hamiltonians of quantum particles under holonomic constraints. We give some examples and discuss some of the related problems.

Rob Waters (University of Bristol), April 14, 2008

The duplicity of zero-one matrices

A matrix with entries in {0,1} can be regarded as a matrix over the integers, or over the field GF(2). We use combinatorial methods to show how much of a difference this distinction can make to its rank, and consider some related questions.

The coupling of stochastic dynamics is a powerful and general probability technique. It is an essential feature of perfect simulation algorithms, which allow to sample exactly from the stationary distribution associated to a (discrete-time) Markov Chain. Order preserving (or monotone) couplings are fundamental to their practical effectiveness. The development of such algorithms by Propp and Wilson in 1996 and their application to the Ising model and other statistical mechanics models is famous. We introduce, formalise and characterise the notions of monotonicity and complete monotonicity for Markov processes with discrete and continuous-time parameter, taking values in a finite partially ordered set. Complete monotonicity is the one required for perfect simulation. We state that the two notions are not equivalent in general. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide. We present and compare results for discrete and continuous-time processes.

Milos Stojakovic (University Novi Sad), march 4, 2008|
Joint work with Dan Hefetz, Michael Krivelevich and Tibor Szabo

For X a finite nonempty set and F a collection of subsets of X, the pair (X, F) is called a positional game on X. It is played by two players Maker and Breaker, where in each move Maker claims one previously unclaimed element of X and then Breaker claims one previously unclaimed element of X. Maker wins if he claims all the elements of a set in F, otherwise Breaker wins.

We study positional games on random graphs. Our main concern is to determine the threshold probability for the existence of Maker's strategy to win the game played on the edges of the random graph G(n,p), for various target families F of winning sets.

We consider random walk on mildly random environment on finite transitive d-regular graphs of increasing girth. After centering and scaling, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d=2: as the limiting object changes from a regular tree to the integers, the noise becomes localized. The talk is based on joint work with Balint Virag.

Mandelbrot's fractal percolation process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0,1]^d in N^d subcubes, and independently retaining or discarding each subcube with probability p or 1-p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d greater than or equal to 2, and in terms of (d-1)-dimensional "sheets" for all d greater than or equal to 3. After introducing the model and some known results, I will present new results concerning the discontinuity of crossing probabilities in all dimensions larger than 2, and the asymptotic behavior of critical values in dimension 2. (Joint work with Erik Broman.)

Given a finite set of lattice points A in Z^d, run a random walk started at the origin until it reaches a point adjacent to the complement of A, and remove that point from A. The internal erosion of A is the random set obtained by iterating this procedure until the origin itself is removed from A. I will present some results about internal erosion in one dimension, and conjectures in higher dimensions.

In recent joint work with Yuval Peres, we have defined a "smash sum" operation which given bounded open sets A and B in R^d, produces a set A\oplus B whose volume is the sum of the volumes of A and B. This operation is intimately related to the scaling limit of internal DLA. After reviewing the construction of the smash sum, I will explain how a process like internal erosion can be used to solve inverse problems such as the following: given a domain A in R^d, find a domain B such that A = B \oplus B.

Alessandro Pellegrinotti (Università degli Studi Roma 3), November 27, 2007

Random walk in fluctuating random environment

A review of the results concerning random walk in fluctuating in time random environment is given. The results concern the validity of the central limit theorem and the time behaviour of correlations.

This is work in progress joint with Frank den Hollander and Gregory Maillard. We study the long-time behavior of the heat equation on $\mathbb Z^d$ driven by a catalytic random potential. The solution of this equation describes the evolution of a ``reactant'' under the influence of a ``catalyst''. In this talk the focus is on the case where the catalyst is modeled by a voter dynamics with opinions 0 and 1 and with a symmetric random walk transition kernel starting from either the Bernoulli measure or the equilibrium measure. We consider the annealed Lyapunov exponents, i.e. the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is not strongly transient, but display an interesting dependence on the diffusion coefficent otherwise. The main obstacle is the nonreversibility of the voter dynamics, since this precludes the application of spectral techniques. Here the duality with coalescing random walks is the key to most of our analysis.

We study ergodic behavior of ''signed'' voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed).

We study the Gibbsian character of time-evolved planar rotor systems on

, , in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure . We model the system by interacting Brownian diffusions moving on circles. We prove that for small times and both arbitrary initial Gibbs measures and arbitrary temperature dynamics, or for long times and both high- or infinite-temperature initial measure and dynamics, the evolved measure stays Gibbsian. Furthermore, we show that for a low-temperature initial measures evolving under infinite-temperature dynamics there is a time interval such that fails to beGibbsian.

Francesco Caravenna (Università degli Studi di Padova), October 23, 2007
(Joint work with Erwin Bolthausen and Béatrice de Tilière)

We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed.

We consider the conditional large deviation behaviour of the sequence of words obtained by cutting an i.i.d. sequence of letters according to a renewal process when fixing a typical letter sequence, and discuss potential applications to stochastic systems in random media. This is joint work in progress with Andreas Greven and Frank den Hollander.

Michael Steele (Wharton University of Pennsylvania), August 20, 2007

Probability Inequalities: Concentration via Local Changes

This talk will provide an introduction to rich and useful class of probability inequalities the essence of which is that if you have a function Z of independent random variables that does not change "too much" if one of the arguments is replaced by an indepentdent copy then Z is itself "concentrated"--- e.g. Z has small variance or thin tails. We will also consider applications of these inequalities to random variables that are of interest in combinatorics and machine learning.

Julien Berestycki (Aix-Marseille 1 and Paris 6), August 14, 2007
Joint work with Nathanael Berestycki and Vlada Limic)

In this talk we prove an asymptotic formula for the number of blocks of a Lambda-coalescent at small times. This formula is a consequence of a more general connection that we obtain between the small-time behavior of Lambda-coalescents on the one hand, and branching processes and continuous random trees on the other hand. This point of view connects in a unified way several recent results of Bertoin and Le Gall, Birkner et al., and Berestycki et al.

Mixing times and diophantine approximations

We study a simple shuffle where the top card of a deck is moved either to the bottom or to position $k$. It turns out that the relaxation time for the location of a card depends in a non-trivial way on the diophantine approximations to k/n.

Invasion percolation is a dynamic process closely linked to critical percolation, but without an external parameter. In joint work with Omer Angel, Frank den Hollander and Gord Slade we showed that the cluster of invaded points consists of a backbone together with sub-critical trees hanging off the backbone, with parameters that increase towards criticality.
By this analysis we derive scaling estimates for the r-point functions and related quantities, and show that the scaling behaviour differs from the incipient infinite cluster. In ongoing joint work with Mathieu Merle, we also analyze the continuum limit of the random trees obtained in invasion percolation

Disordered systems are statistical mechanics models with interactions chosen at random with a given probability distribution. In high temperature finite range systems, in which the interactions can be arbitrarily large with finite probability, the problem of the Griffiths' phase is recovered. By using a multiscale cluster expansion it is possible to prove the exponential decay of correlations in the whole complete analiticity region of the unperturbed model.

Results on the semi-infinite Ising model

Using an extension of Pirogov-Sinai theory, which allow for a countable number of phases with interacting contours, the semi-infinite Ising at small temperature is presented as a many phase contour model. We determine the piecewise analytical structure of the surface free energy and the phase diagram is found. In the latter point we use the correlation inequalities of Lebowitz and Griffiths.

I will give a brief and non-rigorous introduction to lattice trees (a model for branched polymers in statistical physics), super-Brownian motion (a process whose value at time t is a [random] measure), the lace expansion (roughly speaking an inclusion-exclusion analysis of an indicator function) and the connection between them when d>8.

In this talk I will discuss a random graph model based on geometric preferential attachment.

In the course of the talk I will introduce this model by means of other preferential attachment models. We will mainly consider the power-law exponent of the degree sequence, i.e. the fraction of nodes with degree is proportional to .

Ramsey Theory reassures us that the complete disorder is impossible. In graph theory setting this means that for a given a graph $G$ and an integer $k>1$, in any coloring of the edges of the complete graph $K_N$ by $k$ colors, there exists a monochromatic copy of $G$ provided $N$ is large. The smallest integer $N$ with this property is called the Ramsey number $r_k(G)$.

In the first half of my talk, we will briefly survey the most interesting results when $k=2$. In the second half, we look at the case when $k>2$. Here we do not know much even if $G$ is a very simple graph, e.g., a cycle.
Abstract: Ramsey Theory reassures us that the complete disorder is impossible. In graph theory setting this means that for a given a graph $G$ and an integer $k>1$, in any coloring of the edges of the complete graph $K_N$ by $k$ colors, there exists a monochromatic copy of $G$ provided $N$ is large. The smallest integer $N$ with this property is called the Ramsey number $r_k(G)$.

In the first half of my talk, we will briefly survey the most interesting results when $k=2$. In the second half, we look at the case when $k>2$. Here we do not know much even if $G$ is a very simple graph, e.g., a cycle.

Nicolas Pétrélis (EURANDOM), May 1, 2007
(Joint work with Frank den Hollander)

In this talk we will present some results that we obtained with Frank den Hollander about a model of copolymer in an emulsion. This model was introduced by F. den Hollander and S. Whittington in their paper “Diffusion of a heteropolymer in a multi interface medium”. We will focus on the super-critical case (when one of the two types of droplets percolates), and more particularly on the phase transition between full-delocalization in the infinite cluster and partial-localization at the interface between the infinite cluster and the other solvent. We will see that the order of the transition is exactly 2.

The classical ballot theorem states that for a symmetric simple random walk S_1,...,S_n, given an integer k with 0 < k < n, if k and n are of the same parity then P(S_i > 0 for all 0 < i < n | S_n=k) = k/n. We show that essentially the same result holds for any random walk S with mean-zero step size X in the range of attraction of the normal distribution, as long as k is O(sqrt(n)). We also show that this result is essentially best possible. This is joint work with Bruce Reed

It is a safe---albeit imprecise---conjecture that most Lorentz gases in 2D are recurrent. We formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which i.i.d.\ random scatterers are placed in each cell of a co-compact lattice in the plane. We give results towards showing that recurrence is an almost sure property (topological typicality and a 0-1 law that holds in every dimension). The mathematical machinery, including an application of a beautiful theorem by Schmidt and Conze, can be pushed further in the case of simpler systems having the same structure. These include the so-called persistent random walks in random environment. For a large class of them we prove almost sure recurrence.

Dimitris Cheliotis (Bahen Center for Information Technology, Toronto, Canada), April 17, 2007
Joint work with Balint Virag

Repeating patterns for one-dimensional random walk in random environment. A law of the iterated logarithm.

We take a random walk (or diffusion) in a random one-dimensional environment, and we look at its graph at different, increasing scales natural for it. What are the patterns that we will see appear repeatedly? This is a classical problem in the theory of stochastic processes.
Surprisingly, despite the complexity of our process, there is a neat characterization of the repeating patterns.
The analogous result for random walk in a flat, deterministic environment is the well known functional law of the iterated logarithm of Strassen.

The first half of the talk will be introductory. We will describe the model and state some fundamental results for it.

Evsey Morozov (Institute of Applied Mathematical Research and Petrozavodsk University, Russia), April 17, 2007

Stability analysis of a multiserver system with a dependence between workload, input and service time

Including various dependencies in a queueing model to reflect real-life effects, makes it more realistic. Our study concerns a general multiserver queue allowing dependence of the service time of an arriving customer and next interarrival period on both the current waiting time and the server assigned to the arriving customer. There are essentially no assumptions on the (conditional) distributions describing the model. We combine the Markov property of the workload process (to describe dependencies in the framework of Lindley-type recursion) with the regenerative property of that process to study stability using a characterization of the limit behaviour of the renewal process of regenerations. For this system, we establish sufficient stability conditions which are close to being necessary.

We consider the t-improper chromatic number of the Erdos-Renyi random graph. As usual, G(n,p) denotes a random graph with vertex set [n] = {1,...,n} in which the edges are included independently at random with probability p. The t-improper chromatic number chi^t(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we show that chi^t(G(n,p)) is likely to be close to min{np/t, chi(G(n,p))}, as long as t(n) = o(log n) or t(n) = omega(log n). For the case t(n) = Theta(log n), we outline a conjecture for the asymptotic value of chi^t(G(n,p)) which is motivated by large deviations estimates. This is joint work with Colin McDiarmid.

In this talk I will explain some connections between partitions and compositions and their limit shapes when their weight and length grow in a certain regime. More precisely, we consider a uniform measure on partitions of weight n and length m. For n and m growing to infinity with

approximately m! times more compositions than partitions. It implies that many properties of the uniform measures are asymptotically the same for compositions and partitions growing in this regime.

I will show that certain properties of the uniform measures still coinside asymptotically in different grow regime, namely if m^2 = o(n). This will be explained via a notion of sliced Young diagrams.

I'll explain some aspects of the proof of the individual invariance principle for random walks on a percolation cluster from my paper with A. Piatnitski, regarding in particular the use of 2-scale convergence.

Then I'll mention more recent developments on invariance principles for random walks with random conductances.

Using the framework of the hydrodynamic limits, we will discuss the large deviations of the heat current through a diffusive system maintained off equilibrium by two heat baths at unequal temperatures. In particular, we will explain the occurence of a dynamical phase transition which may occur for some models.

The Metropolis algorithm is a basic tool of scientific computing. Useful analysis of the algorithm lies far in the future. I will explain the algorithm through a cryptography application, show how it is characterized as an L1 projection, and present some examples where the running time can be estimated. This last uses techniques of Micro-local analysis and is joint work with Gilles Lebeau.

Abstract For site percolation on the hypercubic lattice, a pattern is a prescribed configuration in a cube of fixed diameter. We show that such patterns occur with positive density on large percolation clusters, and that two distinct patterns must occur in a given ratio. These results are used to prove the ratio limit theorem for percolation, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges. For supercritical percolation, we obtain a slightly stronger result. Generalisations of these methods to other Markov random fields are also discussed.

Consider a lamplighter that randomly walks along a street and that randomly switches on or off the lampposts that stand at every crossing. In this way, one constructs a random process on the set of all possible configurations of 0 and 1 on the integer line (the lampposts) times the integer line it self (the position of the lamplighter). In the last years mathematicians have been interested in such stimulating model. I would like to presents two joint works with W.Woess, on this subject. We associate to the random walk of the lamplighter a random walk on a suitable graph, called Diestel-Leader graph, that is the horocyclic product of two trees. Thank to this approach, we obtain a deeper understanding of the geometry of the lamplighter walk and we are able to deduce the precise asymptotic behavior of the Green kernel and a complete description of the harmonic functions.

Pierluigi Contucci (Università Bologna), February 6, 2007
Joint work with Joel Lebowitz.

A correlation type inequality for spin systems with quenched symmetric random interactions is illustrated and proved. Monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models is derived. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. Conjectured inequalities are discussed.

Christof Kuelske (University of Groningen), February 5, 2007
Joint with Arnaud Le Ny

Consider the ordinary Curie-Weiss Ising model under the site-wise independent spin-flip dynamics which flips the spins symmetrically between plus and minus at constant rate one. This seemingly trivial system shows some peculiar transition phenomena as a function of time when one looks at it on the level of continuity of conditional probabilities (mean-field non-Gibbsianness.) The analysis of the problem is related to an analytical bifurcation phenomenon that can be treated to a great level of concreteness, arising e.g. in explicit parametrizations of critical curves in parameter place. Apart from the interest in the concrete problem we use this example to lay out some basic bifurcation theory ("catastrophe theory") that is useful for studies of phase transitions in mean-field models.

We give a general description of the broken replica symmetry bounds for the free energy in spin glass theory. We show how to extend them to the case of many coupled replicas. In this frame, we discuss the phenomenon of spontaneous replica symmetry breaking, discovered by Giorgio Parisi.
Our presentation will be completely elementary and self-contained.

An introduction to renormalisation and discretisation techniques for continuum percolation models

Renormalisation and discretisation are two powerful techniques for studying spatial stochastic systems. In brief, renormalisation is analysing a model on a different scale and discretisation is when a grid is imposed on a continuous space. Although often conceptually simple, examples of these techniques in the literature tend to be difficult to read.

These techniques will be illustrated by examples from a new continuum percolation model. Continuum percolation is the study of the connected components of random graphs, where the vertices of the graph live on a continuous space. The talk should be accessible for a general mathematical audience, with a number of open problems being presented that should interest specialists in the field. This talk presents joint work with Misja Nuyens and is based on a recently submitted paper available at www.cs.vu.nl/~ajg/preprints.ph

Francesca Nardi (Technische Universiteit Eindhoven), Mini-course November, 28, December 5 and 19, 2006

Lecture of 28 November 2006 “ Metastability for Ising model with the Glauber dynamics” Abstract: we consider the ferromagnetic Ising model on finite volume, for large values of the parameter beta, the measure will be concentrated around the absolute minima of H. We consider Glauber dynamics. A very interesting problem is the study of the so called "metastable behavior", namely the mechanism which, starting from a local minimum of H (metastable state), leads the system, via a large deviation, to the state with minimum energy (stable state). This problem is important for studying theoretical models of metastability in statistical mechanics. From the probabilistic point of view, metastable decay is a first exit problem from a suitable domain in the configuration space, for a reversible Markov chain in which those transitions which increase the energy H are exponentially small in the parameter beta. In the limit beta going to infinite, this problem can be treated in great generality within the Freidlin and Wentzel theory of small random perturbations of dynamical systems (see[1]). We will compare the results of the pioneer papers of Neves and Schonmann (see [2] and [3]) with the more recent results obtained by Bovier and Manzo (see [4]) and Manzo Nardi Olivieri Scoppola (see [5]).

Lecture of 5 December 2006 “ Metastability for anisotropic Ising model with the Glauber dynamics” Abstract: we consider the anisotropic ferromagnetic Ising model on finite volume, for large values of the parameter beta evolving under Glauber dynamics. We study metastability in this framework for which the equilibrium Wulff shape is not a cube. Considering large finite volumes, small magnetic fields, and very low temperatures, we show that the typical paths in the transition from the metastable to the stable phase are through sequences of ‘non-Wulff’ configurations. These results are obtained by Koteck\'y and Olivieri in [6].

Lecture of 19 December 2006 “Metastability and nucleation for conservative dynamics.” We analyze metastability and nucleation in the context of a “local version” of the Kawasaki dynamics for the two-dimensional Ising lattice gas in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. Let Lambda be a sufficiently large finite box in the two dimensional lattice. Particles perform simple exclusion on Lambda, but when they occupy neighboring sites they feel a binding energy -U < 0. Along each bond touching the boundary of Lambda from the outside, particles are created with rate rho=exp(-Delta beta) and are annihilated with rate 1, where beta is the inverse temperature and rho > 0 is an activity parameter. Thus, the boundary of Lambda plays the role of an infinite gas reservoir with density rho. We take Delta in the interval (U, 2U) where the totally empty (full) configuration can be naturally associated to metastability (stability). We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. We will compare the results by den Hollander, Olivieri and Scoppola in [7] using pathwise approach and results by Bovier, den Hollander and Nardi in [8] that combine geometric and potential theoretic arguments. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet are highly sensitive to this motion.

[1] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer-Verlag, 1984). [2] E. J. Neves and R. H. Schonmann, Critical droplets and metastability for a Glauber dynamics at very low temperature, Comm. Math. Phys. 137:209–230 (1991). [3] E. J. Neves and R. H. Schonmann, Behaviour of droplets for a class of Glauber dynamics at very low temperatures, Probab. Theory Related Fields 91:331–354 (1992). [4] A. Bovier and F. Manzo, Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics, J. Stat. Phys. 107:757–779 (2002). [5] F. Manzo, F.R. Nardi, E. Olivieri, E. Scoppola,``On the essential features of metastability: tunnelling time and critical configurations." Journal of Statistical Physics Vol. 115, (2004), 591-642. [6] Koteck\'y, R.; Olivieri, E. Stochastic models for nucleation and crystal growth. Probabilistic methods in mathematical physics (Siena, 1991), 264-275, World Sci. Publ., River Edge, NJ, 1992. 82C44. [7] F. den Hollander, E. Olivieri and E. Scoppola, Metastability and nucleation for conservative dynamics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. Vol. 41, page 1424--1498 (2000). [8] A. Bovier, F. den Hollander, F.R. Nardi, ``Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary." Probability Theory and Related Fields, Vol 135, 265-310 (2006).

We couple a lattice gas of labelled particles, with low density $\rho$ evolving under Kawasaki dynamics inside a large two-dimensional box, with a gas of of independent random walks.

We compare the trajectories of the particles in the two dynamics and show that, under some general hypotheses on the initial configuration and at inverse temperature not higher than $-cst\ln\rho$, the two dynamics essentially behave in the same way up to times much larger than $1/\rho$.

In particular we derive some estimates for the probability of seeing at time $t$ a set of given particles in a given set of sites which are similar to those obtained for independent random walks.

Gregory Maillard (EPFL Lausanne, Switzerland), November 23, 2006
Joint work with Roberto Fernandez

We are interested in the study of phase transitions for continuous, positive and non-stationary chains with complete connections. We use the Hierarchical model introduced by Dyson (1969--1972) in his seminal work on phase transition for the one-dimensional Ising model to build, in a very simple way, a one-sided version of a Hierarchical model for which the corresponding chain with complete connections does not satisfy uniqueness.

We will show why this result is complementary to those existing in the literature and discuss the optimality of uniqueness criteria.

This project concerns two aspects of the simulation of fracture. First analysis is done of the energy distribution within a lattice at different points during the fracture process by calculating several of its moments for different lattice sizes. The energy moments are hypothesized to exhibit multifractal scaling with the lattice size. Only for the point right before mechanical breakdown of the system, a deviation from monofractality is concluded.

Another aspect builds on the conclusion that random percolation can model strong-disorder fracture with respect to several properties that exhibit fractality. Analysis is done whether modifying random percolation into gradient percolation based on the damage profile from the fracture simulation can improve the similarity to strong-disorder fracture with respect to these properties. It is concluded that gradient percolation based on fracture's damage profile provides little improvement, but surprisingly, exaggerating this profile does improve the similarity.

We present a model for artificial cells and accompanying simulation results. Some important questions are raised with respect to possible analytical approaches to understand these results and the audience is invited to take active part in potential pathways to follow.

We introduce an interacting spin model to describe the output of a contact between two populations carrying different cultural attitudes. The problem of establishing the presence of abrupt swings is reduced to the study of the phase transitions of the model.

We discuss a relation between generalised random graphs (see e.g. Britton, Deijfen and Martin-Löf (to appear in J. Statist. Phys.)) and an

(Susceptible

Infectious

Removed/immune) epidemic model on a directed (possibly complete) network

. In the epidemic model the vertex set

stands for the individuals and the edge set

stands for the set of connections between individuals.

Our generalised random graph model is described as follows: Every vertex (individual) has a 2-dimensional random vector

assigned to it all distributed as the random varioable

. An edge from

is open with probability

. and otherwise it is closed. Conditioned on the random vectors

We interpret this model in a epidemiological setting as follows: If there is an open edge from

, then at least one infectious contact is made from

. i.e. if

becomes infected itself, then

will be infected as well, if it has not been infected before.

We compare different “epidemics” with given expected

and

and show that if

and

are independent, then the process with fixed

and

is a worst case scenario, in the sense that the probability of a large outbreak as well as the expected number of ultimately removed individuals is maximal.To obtain this result we use an idea of Kuulasmaa (J. Appl. Probab. 1982).

We study the rotor router model and two deterministic sandpile models. For the rotor router model in , Levine and Peres have proven that the limiting shape of the growth cluster is a ball. For the other two models, only some bounds in dimension 2 are known. A unified approach for these models with a new parameter , playing the role of the initial number of particles at each site, allows to prove a number of new limiting shape results in any dimension .

For the rotor router model, the limiting shape is a ball for all values of . For one of the sandpile models, and (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a ball in the limit . Finally, we prove that the rotor router shape contains a diamond, which is a new result even in the case studied by Levine and Peres.

We consider the problem of metastability for a stochastic dynamics with parallel updating rule and we study the exit from the metastable state evaluating the exit time and the typical exit path. Moreover we give sharp estimates on this exit time.

A random geometric graph is constructed by taking vertices at random (i.i.d. according to some probability distribution ) and including an edge between and if where .

We prove a conjecture of Penrose stating that when then the probability distribution of the clique number becomes concentrated on two consecutive integers in the sense that

for some sequence .

We also show that the same holds for a number of other graph parameters including the chromatic number .

A series of celebrated results establish that a similar phenomenon occurs in the Erdős-Rényi or -model of random graphs.

Recently there has been a lot of interest in the use of random graphs as models for complex networks. This has inspired a number of models for generating random graphs with prescribed degree distribution. A natural generalization of the problem of generating random graphs with given degree distribution is to consider spatial versions of the same problem, where geometric aspects play a role. I will describe results on Z.

Francesco Caravenna (Institute of Mathematics, University of Zurich), July 13, 2006
Joint work with J.-D. Deuschel

We consider a random field $\phi: N -> R$ with Laplacian interactions of the form $V(\Delta\phi)$, for a large class of potentials $V$, and with in addition a delta-pinning reward for the field to touch the x-axis, that plays the role of a defect line. Denoting by $\epsilon \ge 0$ the intensity of the pinning reward, we show that there is a phase transition at $\epsilon = \epsilon_c > 0$ between a delocalized regime $(\epsilon \le \epsilon_c)$, in which the field wanders away from the defect line, and a localized regime $(\epsilon > \epsilon_c)$, in which the field sticks close to it. Using an approach based on renewal theory we extract the scaling limits of the model. In particular, we show that in the critical regime $(\epsilon = \epsilon_c)$ the rescaled field converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5.

Applications of probability theory to polymers have been numerous. Starting with Flory, who realised that polymers in good solvents can be described as self-avoiding random walks, probability theory has shown to be important in the investigation of a variety of problems in polymer physics. In this seminar I will present an analysis of so-called multilayer Markov chains and apply the results to a model of a tethered polymer chain in shear flow. It is found that the stationary probability measure in the direction of the flow is nonmonotonic, and has several maxima and minima for sufficiently high shear rates. This is in agreement with the experimental observation of "cyclic dynamics" for such polymer systems. Estimates for the stationary variance and expectation value were obtained and showed to be in accordance with our numerical results.

I will present a method introduced by Vdovichenko (1965) to compute the partition sum of the two-dimensional Ising model. The main idea of the method is to replace the sum over contour diagrams that represent the interfaces between the Ising spins by a sum over loop diagrams. This representation has the nice property that the contribution from all loop diagrams with s loops is just a product of s single-loop contributions. The single loops are generated by a transition matrix whose eigenvalues determine the single-loop contribution and thus the partition sum.

The fact that factorization into single-loop contributions takes place implies that to study a single Vdovichenko loop, we may ignore all the others. This raises the question whether one can use this approach to study properties of a single interface in the Ising model (e.g. the interface whose conjectured scaling limit at the critical point is SLE).

We present the following model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire, there must be exactly one bead on each neighbouring wire.

We construct a one-parameter family of "uniform" Gibbs measures on the bead configurations. When endowed with one of these measures, this model present connections with the GUE ensemble. We explain then that this process is the limit of any dimer model on a planar bipartite graph when some weights degenerate.

The problem of entrance time in small sets is one of the oldest question in ergodic theory motivated by statistical mechanics (Boltzmann). We will review some of the results for dynamical systems, and the extensions to repetition times, statistics of extreme and quasi invariant measures. If time permits we will sketch a new proof of the exponential law based on an idea of Kolmogorov for proof of the CLT.

Exponential inequalities can be seen as rough large deviations upper bounds for observables which are not sums of random variables. Devroye inequalities are the corresponding variance estimates. We will consider the case of dependent random variables (non product measure) corresponding to dynamical systems and discuss the consequences for concentration and some applications. We will briefly sketch a proof based on a coupling argument.

Markus Heydenreich (EURANDOM/TUe), April 11, 2006
Joint work with Remco van der Hofstad

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation.

We use a relatively simple coupling argument to show that, up to logarithmic corrections, this largest critical cluster scales like V^{2/3}, where V is the volume of the torus.

Interestingly, this is the same asymptotic behavior that Erds and Reni (1960) observed for the critical random graph, which is the special case of percolation on the complete graph.

Our results establish a conjecture by Aizenman from 1997, apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. It turns out that scaling of the largest connected component under bulk boundary conditions, as studied by Aizenman (1997), is quite different from the scaling under periodic boundary conditions.

Our method makes crucial use of the results by Borgs, Chayes, van der Hofstad, Slade & Spencer (2005), where partial results of V^{2/3} scaling on high-dimensional tori were proved.

The report is devoted to the stochastic flows on R^d which consists of coalescing diffusion particles. The new type of the spatial stochastic integral with respect to such flow is introduced. Based on this construction the Girsanov theorem for the Arratia flow is proved. The Clark representation for the functionals from Arratia flow also is obtained. The investigation of the spatial properties of Arratia flow leads to the boundary value problems for the stochastic anticipating equations. The solutions to these problems are proposed.

Starting from the rigorous picture of the full scaling limit of critical site percolation on the triangular lattice, obtained in collaboration with C. M. Newman, I will discuss some mostly heuristic and conjectural new developments concerning the near-critical scaling limit of 2D percolation and related model (such as the minimal spanning tree) obtained in collaboration with L. R. Fontes and C. M. Newman. These include a type of conformal covariance that replaces the full conformal invariance typical of systems at the critical point.

The Internal Diffusion Limited Aggregation has been introduced by Diaconis and Fulton in 1991. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We present a shape theorem for the Internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model around its limiting shape.

Stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy are investigated. These networks are motivated by bandwidth allocation problems in wireless networks. The associated
(non-reversible) Markov jump processes are analyzed under a thermodynamic limit regime, i.e. with symmetry properties and when the number of nodes goes to infinity. Under some conditions on the parameters, a metastability property is proved. It is shown that, despite the fact that the dynamic of these networks is local, several equilibrium points coexist in the limit. The implications of this unusual property (for queueing networks) are discussed. The key ingredient of the proof of this result is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their corresponding local minima and saddle points. Cases with a unique equilibrium point are also analyzed.

We introduce conditions that imply, in the multidimensional setting, a strong law of large numbers with non-vanishing limiting velocity (which we refer to as ballistic behavior) and an invariance principle with non-degenerate covariance matrix. As an application of our results, we give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides new examples of ballistic diffusions. With our methods, we are further able to rederive the well-known ballistic character of a class of diffusions in random environment with a divergence-free drift by simply checking the above mentioned sufficient condition for ballistic behavior.

Jan Swart (UTIA, Academy of Sciences of the Czech Republic), December 16, 2005

In this talk, we will consider contact processes on general countable groups, which includes the usual d-dimensional integer lattice, regular trees, and more. We will see that the expected number of sites has a well-defined exponential growth rate, which is closely tied to how the process looks as seen from a typical (`Palmed') infected site at a typical late time. In particular, if the process grows subexponentially, as is for example the case on the usual integer lattice, we will prove that the process as seen from a typical infected site converges, as time tends to infinity, to the upper invariant measure conditioned on the origin being infected.

Jan Swart (UTIA, Academy of Sciences of the Czech Republic), December 9, 2005

Systems of linearly interacting diffusions indexed by the hierarchical group can be analyzed by means of a renormalization transformation, which tells one how the local laws defining the process are related to its behavior on large space and time scales. In the case where the single components are one-dimensional, this technique has been developed by Dawson, Greven, and others, who showed that the behavior of such systems is highly universal, in the sense that the behavior on large space and time scales is independent of the local laws. In the case where the single components are multi-dimensional, a much richer universality structure is expected, but here there are still a lot of open problems. In this talk I will give a general overview on renormalization of linearly interacting diffusions, with special attention to two-dimensional catalyctic Wright-Fisher diffusions, which were recently treated by Klaus Fleischmann and me.

Andrea Collevecchio (University D'Annunzio, Pescara, Italy), November 18, 2005

We introduce a simple technique to prove the transience of processes defined on trees generated by branching processes. We apply this method to once-reinforced random walk and vertex-reinforced jump process.

I will discuss a stochastic model, describing the spread of infection on (social) networks. The main properties of networks that has to be taken into account are the number of individuals, the distribution of the number of neighbors and the number of short loops in the network. The number of short loops is important because some of the contacts of infective individuals may be with individuals that are no longer susceptible. Short loops may arise naturally in social networks because of ``the friends of my friends are also my friends''.

In epidemic literature the deterministic model of pair approximations is proposed to analyse epidemics on networks. Because this model is deterministic it is not possible to use it for estimating the probability of extinction of the infection. I will spend a few words on pair approximations and give some drawbacks.

Another method proposed is approximating the network with random graphs. On these random graphs it is rather easy to describe the spread of an infection and to estimate the probability of extinction of the infection. However the proposed random graphs in literature do not contain small loops in it.

In this talk a construction of random graphs is explained, that do have the same degree distribution and the same number of loops of length three as the original network. By this construction the strong features of pair approximations and random graph methods are combined.

Lace expansion for the Ising model II. Bounds on diagrams

The lace expansion has been a powerful tool to investigate mean-field behavior for various stochastic-geometrical models. Recently, the lace expansion for the Ising model has been proved for the first time. In the previous talk, I explained what is the lace expansion for the Ising model, as well as its consequence assuming bounds on the expansion coefficients.

In this talk, we continue the derivation of the expansion. One of the key points for the derivation is the source-switching lemma, which was first discovered by Griffiths, Hurst and Sherman, and then developed by Aizenman. I will show how it is used to complete the expansion and how it is extended to prove diagrammatic bounds on the expansion coefficients. These diagrammatic bounds are optimal to prove the mean-field behavior above 4 dimensions (with sufficiently large coordination number).

Leandro P. R. Pimentel (EPFL, Laussane), September 29, 2005
This is a joint work with P. A. Ferrari and J. B. Martin

We study the \emph{competition interface} between two growing clusters for a simple growth model (\emph{last-passage percolation}) in a bidimensional sector. Using technology built up for geodesics in percolation and a relation with the totally asymmetric simple exclusion process, we show that the asymptotic inclination and the fluctuations of this interface depends on the geometry of the initial configuration.

We will show a proof of the Ghirlanda-Guerra identities which only requires that the variance of the Hamiltonian grows like the volume (thermodynamic stability). Our result is expressed in terms of the model's covariance and applies to all Gaussian spin glass models.

Malwina Luczak (London School of Economics), September 14, 2005
Joint work with Svante Janson.

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. We let $n \to \infty$. Then, under some regularity conditions on the degree sequence, we give conditions on its asymptotic shape that imply that with high probability the $k$-core is empty; and other conditions that imply that with highprobability the $k$-core is non-empty and the sizes of its vertex and edge sets satisfy a law of large numbers. Under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer and Wormald on existence and size of a $k$-core in $G(n,p)$ and $G(n,m)$.
Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

I will show how to generate a finite range decomposition for the covariance of a gaussian measure on a lattice. And I will use such a decomposition to study the RG transformation for a generic polymer system with a weak perturbation to the gaussian measure.

The path space measure for a (non-equilibrium) stochastic process $P_{\Phi}$ may be formally related to another (equilibrium) pathspace measure $P_{0}$ through the "stochastic action"
\[
P_{\Phi} = \exp(\Delta\aleph_{\Phi,0}) P_{0}, \] where $\Delta\aleph_{\Phi,0}$ is the difference in stochastic actions between the two processes, and $\Phi$ is the so-called field that parametrizes the one process with respect to the other.

The (time-)antisymmetric part of this action is related to the entropy production and through it to the Galavotti-Cohen fluctuation relation.
The question naturally rises: Then what is the meaning and significance of the time-{\em symmetric} part?

Knowledge and understanding of the time-symmetric part is crucial for systems that are far from equilibrium (beyond the linear regime), such as biological systems. We introduce the notion of field-reversal (an analogue of time-reversal) in order to study its properties. With help of some examples and by seeking analogy with other results we will share some of the insight that we have recently obtained.

In this talk we consider long range percolation models on Z^d. For various families of non-summable edge probabilities (p_n), we show that there exists an integer K, such that the truncated model, in which all edges whose sizes is larger than K are suppressed has a infinite cluster almost surely.

Karl Petersen (University of North Carolina Chapel Hill, NC, USA), June 24, 2005

A one-step shift of finite type is among the simplest of dynamical systems to describe, and a one-block factor map is among the simplest models of information loss through coding. Yet this setup involves surprising complexity and leaves open many natural questions, some of which have practical implications for information handling. The difficulties begin to emerge when one realizes that the image of a one-step Markov measure is probably no longer Markov, and its entropy can be hard to determine. In recent work with Anthony Quas and Sujin Shin, we considered a relative Shannon-Parry property: does every ergodic measure on the image have a unique preimage of maximal entropy? The answer is no, but we showed that the number of maximal entropy lifts is at least finite. Key tools in this area are the ideas of compensation function (introduced by Boyle and Tuncel and developed by Walters) and relative pressure and equilibrium states (Ledrappier and Walters). In a paper with Shin that is in press, we compared two natural definitions of relative pressure and showed that they are almost equivalent. Related results of Shin have implications for the identification of measures of maximal Hausdorff dimension for the restrictions of expanding maps on manifolds to compact invariant sets.

Recently there has been a lot of attention on random graph models with an arbitrary prescribed degree distribution. In particular, models with power law degree distributions have been studied. In this talk, we will consider the problem of generating a random graph with a prescribed degree distribution under the extra restriction that the graph should be simple, that is, it cannot contain any self-loops or multiple edges between vertices. A number of possible algorithms will be described and they are all shown to give the correct degree distribution in the limit of large graph size. If time permits, a spatial version of the problem of generating random graphs with prescribed degree distribution will also be discussed.

Hidden symmetry of Kramers' equation and the problem of finding reaction paths

Kramers' equation can be extended in a way that reveals an underlying (super) symmetry. I will show how one can use this ideas as a basis for theoretical and practical methods for the study of reaction currents in phase-space (a central problem in Physical Chemistry). The same methods can be used to reveal separatrices and resonant tori in Hamiltonian systems.

In this talk I will present results on a random graph with nodes, where node has degree and are i.i.d. with . Our main assumption is that for some , and where is slowly varying at infinity. The graph model is a variant of the so-called configuration model.

The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when . We prove that for the graph distance grows like , where the base of the logarithm equals . This confirms the heuristic argument of Newman, Strogatz and Watts. In addition we characterize the asymptotics of the random fluctuations around .

For and under some additional technical assumption, we prove that the graph distance grows like . Again we are able to characterize the asymptotics of the random fluctuations around this mean.

Finally for , the graph distance is concentrated on the values and , as .

For , the talk is based on a paper written jointly with Remco van der Hofstad and Piet Van Mieghem. The other cases are based on two papers together with Remco van der Hofstad and Dmitri Znamenski. A survey article, that treats all three regions for can be downloaded from the website:

http://ssor.twi.tudelft.nl/~gerardh/

We study the asymptotic behaviour of the probability that a Brownian motion in R^m ( m > 2 ), starting at x , hits a non polar compact set K before large time t .

Reinforced random walk (RRW) is a broad class of processes which jump between nearest neighbor vertices of graphs, and prefer visiting often visited ones over seldom visited ones. These processes can be used to model phenomena with a "nostalgic" component. We will describe the behaviour of RRWs on certain graphs, in particular trees.

In this talk we will consider a discrete random pinning model with an entropic repulsion. We will be particularly interested in the convergence at high temperature of this model toward a continuous one. This continuous model brings us important informations on the discrete one.

European Institute for Statistics, Probability, Stochastic Operations Research and its Applications