
Stochastics 
Theoretical and
Applied Research

2011 November 3, 16.00h  17.00h.: Sandeep Juneja, visitor Mark Kac seminar 2010 Stochastic flows by Andrey Dorogovtsev Mia Deijfen (Stockholm University) Thomas Gerds (University of Copenhagen) June 10,2011 Mark Kac
seminar Utrecht, Kromme Nieuwegracht 80,
room 130 !! 11:1513:00 14:3016:15 Title : The deformed exponential function F(x,y) = sum_{n=0}^infty x^n y^{n(n1)/2} / n! and a plethora of related things Abstract morning talk Reference: Abstract afternoon talk References: May 24, 2011
STAR seminar Program May 13, 2011 Mark Kac
seminar 14:3016:15 Abstract JeanRené Chazottes We report on recent positive and negative results for the convergence of Gibbs measures as temperature goes to zero. The Gibbs measures considered are on {0,1}^{Z^d}. Abstract Pierre Nolin For twodimensional independent percolation, RussoSeymourWelsh (RSW) bounds on crossing probabilities are an important apriori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality. In this talk, we prove RSWtype uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results  including some new ones  among which are the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the halfplane onearm exponent. This is joint work with H. DuminilCopin and C. Hongler. The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac April 1, 2011 Mark Kac seminar Utrecht,
Janskerkhof 15a, room 204 14:3016:15 Abstract Jean Bertoin Knuth's parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking with n sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. It is known from a work by Chassaing and Louchard that the formation of large occupied blocks is governed by the socalled additive coalescent. We incorporate fires to this model by throwing Molotov cocktails on each site at a smaller rate n^{alpha}. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when the size of the parking is large, the parking becomes saturated at a time close to 1 (i.e., as in the absence of fire) for alpha > 2/3, whereas for alpha < 2/3, the mean occupation approaches 1 at time 1 but then quickly drops to 0 before the parking is ever saturated. Abstract Francesca Nardi We study a twodimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to and including a multiplicative factor of order one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived for a certain subregion of the metastable region. The proof involves three modeldependent quantities: the energy, the shape and the number of the critical droplets. The main motivation is to understand metastability of multitype particle systems. It turns out that for two types of particles the geometry of subcritical and critical droplets is more complex than for one type of particle. Consequently, it is a somewhat delicate matter to capture the proper mechanisms behind the growing and the shrinking of subcritical droplets until a critical droplet is formed. The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac March 4, 2011 Mark Kac seminarFifth Our main speaker Alan Sokal will give his first of three lectures on "Some combinatorics and analysis arising out of the Potts model". 11:1513:00 Abstract Alan Sokal The multivariate Tutte polynomial (known to physicists as the Pottsmodel partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar twovariable Tutte polynomial  and therefore also its onevariable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial  but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the YangLee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. Reference: Abstract Anne Fey In the first part of the talk, I will present the parallel chip firing model. In this model, we start with some number of chips on each vertex of a graph, and evolve by parallel firing: in each time step, every vertex that has at least as many chips as neighbors, fires one chip to each neighbor. It has been observed that if we start with the chips uniformly at random placed on a large graph, then the 'density' (average number of chips per vertex) suffices to predict with high accuracy the 'activity' (average number of firing vertices per time step). A plot of the activity as a function of the density reminds one of a staircase. I will show several examples and the proof of our staircase theorem for the flower graph. The second part of the talk will be about a popular theory of selforganized criticality, that relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the driven abelian sandpile model should be equal to the threshold density of the corresponding parallel chip firing model. This "density conjecture" has been proved for the underlying graph Z. Research into this conjecture has focused mainly on the underlying graph Z^2: the stationary density was proved to be equal to 17/8 to at least twelve decimals, while the threshold density was simulated to be 2,125 to three decimals. We have investigated both the driven sandpile model and the parallel chip firing model for several graphs, and we show (by largescale simulation or by proof) that the stationary density is not equal to the threshold density when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. (joint work with Lionel Levine and David Wilson) The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac February 4, 2011 Mark Kac seminar Fourth 11:1513:00 14:3016:15 Abstract Michel Ledoux Logarithmic Sobolev inequalities, introduced by Stam in information theory and Gross in the analysis of infinite dimensional diffusion operators, appear as a main tool in the study of convergence to equilibrium of Markov chains and semigroups in various settings. Their analysis involve a large number of methods, ranging from functional analysis, geometry, optimal transportation, statistical mechanics etc. The first part of the talk will be devoted to a general introduction to the subject of logarithmic Sobolev inequalities. The second part will present some (old and new) results on logarithmic Sobolev inequalities for continuous unbounded spin systems, both in the perturbative and conservative regimes. Abstract Roman Kotecky Recent efforts to link macroscopic nonlinear elasticity with microscopic gradient models at nonzero temperature will be discussed. In particular, we will be concerned with a microscopic derivation of CauchyBorn rule and nonlinear elastic free energies. The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac December 10, 2010 Third Mark Kac seminar 11:1513:00 Speaker: Christian
Maes (KU Leuven) 14:3016:15 Speaker: Frank den
Hollander (Leiden University and EURANDOM) Abstract Christian Maes Fluctuation theory is at the heart of statistical mechanics. We discuss the relevance of the occupation fluctuations in clarifying the nature of entropy production principles, in providing a Liapunov function for the relaxation towards nonequilibrium, and for amending the standard fluctuationdissipation theorem. These basic concepts and results can already be explained for Markov processes with finite state space, but we aim at their relevance for spatially extended systems. [Joint work with many people; see also my webpage for the relevant papers http://itf.fys.kuleuven.be/~christ/] Abstract Frank den Hollander We consider an interacting particle system on the integer lattice in equilibrium, constituting a dynamic random environment, together with a nearestneighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. We describe some recent results for the empirical speed of the walk (law of large numbers, central limit theorem, and large deviation principle) for different choices of the interacting particle system. We compare these results with what is known for static random environments, describe recent extensions, and list open problems. [Based on joint work with Luca Avena, Frank Redig, Renato dos Santos and Vladas Sidoravicius.] The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac November 10, 2010 Stochastic
flows In this talk we give a brief introduction to the theory of stochastic flows and state some actual problems related to this field. In particular, stochastic integration with respect to a flow with singularities will be discussed, a Girsanovtype theorem and a largedeviations principle for flows of Brownian particles will be presented. November 5, 2010 Mark Kac seminar 11:1513:00 14:3016:15 Abstract Stefan Grosskinsky In recent years condensation phenomena in interacting particle systems have attracted a lot of research interest in statistical mechanics and probability. Most results in that direction basically study a family of (grand canonical) product measures for particle configurations, indexed by a parameter that controls the particle density. If the range of this parameter is bounded condensation can occur, i.e., a macroscopic fraction of all the particles concentrates on a single lattice site. Mathematically, this can be related to large deviation properties of heavytailed distributions, to a breakdown of the usual law of large numbers for triangular arrays, or to spatial inhomogeneities leading to different tails of the marginal distributions. All three cases have been studied in the context of zerorange processes, where I will give an overview of the main results and recent developments (including a collaboration with I. Armendariz and M. Loulakis). In recent work with Frank Redig and Kiamars Vafayi we establish condensation also for the inclusion process, which is a 'bosonic' analog of the exclusion process, and for the Brownian energy process, which provides an interesting example with continuous state space. Abstract Sacha Friedli We consider onedimensional stationary processes introduced by Doeblin and Fortet in 1937, specified by a regular gfunction. In the first part we describe the original uniqueness result of DoeblinFortet, under summability of the variation of g, and the more recent result of JohannsonOberg giving uniqueness under square summability. In the second part we will consider the mechanism invented by Bramson and Kalikow, giving nonuniqueness for a class of attractive gmeasures, and discuss a problem related to their example. The Mark Kac
seminar is supported by FOM and is an activity of the STAR stochastics
cluster. See also our website
http://www.win.tue.nl/markkac
October 27, 2010 Seminarium / colloquium kansrekening en statistiek Speaker: Mia Deijfen (Stockholm University) Title: Preferential attachment models and general branching processes When: 11:00 hrs on Wednesday October 27, 2010 Place: TU Delft, Faculty EWI, Mekelweg 4  Snijderszaal (1st floor) Abstract: A much studied type of models for growing networks is based on so called preferential attachement: vertices are succesively added to the network and are attached to existing vertices with probability proportional to degree. This mechanism has been shown to lead to power law degree distributions, which is in agreement with empirical studies on many types of real networks. I shall describe how general branching processes can be used to derive results on the degree distribution in preferential attachment models and also in an extensions of the model where vertices are not just added to the network but may also be removed. October 1, 2010 Mark Kac seminar 11:1513:00 14:3016:15 Abstract Bricmont In the first half of the talk, I will review the general ideas, going back to Boltzmann, on the relationship between microscopic and macroscopic physical laws. In the second half, I will discuss several models of coupled systems for which the derivation of one particular microscopic law, Fourier's law, can be investigated. Abstract Berestycki A series of recent conjectures by B. Derrida, E. Brunet and coauthors concerning the behavior of a class of particle systems related to the noisy (or stochastic) FKPP traveling wave equation have generated a renewed interest in the study of Branching random walks with absorption, We will present some classical and recent results on those models. I will in particular focus on the description of the asymptotic genealogy of those systems, which turn out to be described by the BolthausenSznitman coalescent, a model which is conjectured to be a universal limit in several classes of models in statistical physics. The Mark Kac seminar is supported by FOM and is an activity of the STAR stochastics cluster. See also our website http://www.win.tue.nl/markkac September 27, 2010 Confidence scores for prediction models Speaker: Thomas Gerds (University of Copenhagen) Abstract Machine learning provides many alternative strategies for building a prediction model based on training data. Prediction models are routinely compared by means of their prediction performance in independent validation data. If only one data set is available for training and validation, then rival strategies can still be compared based on repeated splits of the same data (see e.g. [1], [2], [3]). Often however the overall performance of rival strategies is similar and it is thus difficult to decide for one model. Here we investigate the variability of the prediction models that results when the same modelling strategy is applied to different training sets. For each modelling strategy we estimate a confidence score based on the same splits of the data. Population average confidence scores can then be used to distinguish rival prediction models with similar prediction performances. Furthermore, on the subject level a confidence score may provide useful supplementary information for new patients who want to base a medical decision on predicted risk. The ideas are illustrated using examples from medical statistics, also with highdimensional data. * AM Molinaro, R Simon, and RM Pfeiffer. Prediction error estimation: a comparison of resampling. Bioinformatics, 21:33013307, 2005. * TA Gerds, T Cai, and M Schumacher. The performance of risk prediction models. Biometrical Journal, 50(4):457479, 2008. * MA van de Wiel, J Berkhof, and WN van Wieringen. Testing the prediction error difference between two predictors. Biostatistics ,10:550–560, 2009.
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