Submitted titles and abstracts

Tonći Antunović (UCLA): Asymptotics of permanents of certain matrices with heavy tail entries

Abstract: Permanents of random matrices with iid entries of positive mean and finite variance are very well understood. In a more difficult zero mean case Tao and Vu proved a law of large numbers for the logarithm of permanent of Bernoulli matrices, demonstrating a significantly different behavior from the positive mean case. We will present a law of large numbers for the logarithm of permanent of certain random matrices with non-negative heavy-tailed entries, show how the limit can be read from the tail behavior of the entries, and demonstrate the phase transition as the mean of the entries becomes infinite.

Joris Bierkens (Radbound University): Linear equations for optimal control and relations to large deviations theory

Abstract: Consider a general Markov process (i.e. a Markov chain or a diffusion process) which may be controlled, i.e. we may change its dynamical behaviour, usually in order to achieve a certain goal or minimize some optimality criterion. Typically, the optimally controlled process may be obtained through the principle of dynamic programming, which yields the (non-linear) Hamilton-Jacobi-Bellman equations. Under certain, quite broad conditions, such problems may be transformed into linear problems. I will discuss how, associated with the generator of a Markov process, one may define control problems which coincide with the solution of i) the Cauchy problem, ii) the Laplace problem, and iii) the linear eigenvalue problem. Interestingly, in case iii), we obtain in a purely classical setting the time independent Schroedinger equation for the square root of a density function. The correspondence between control problems and the Cauchy problem provides a Monte-Carlo sampling method for the solution of control problems. There is a strong connection with the theory of large deviations which will be discussed.

Tim vande Brug (VU Amsterdam): Fat fractal percolation and k-fractal percolation

Abstract: We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.

Rene Petrus Conijn (VU Amsterdam): The size of the largest clusters in 2D critical percolation

Abstract: For two dimensional critical percolation it is proved by Borgs, Chayes, Kesten and Spencer that the size of the maximal cluster in an n by n box is of order n^2 pi(n). Here pi(n) stands for the probability that the origin is connected with the boundary of the box. We have some recent results on the size of the largest clusters. Partly joint work with Rob van den Berg and with Federico Camia and Demeter Kiss

Stephen DeSalvo (UCLA): Probabilistic Divide-and-Conquer -- A new method for random sampling with Integer Partitions as an example

Abstract: New method for random sampling, works very well for integer partitions. Applications to other structures, plus simple trick significantly improves random sampling algorithms, with simple trick relying on understanding the underlying combinatorial structure.

Lisa Hartung (Bonn): The extremal process of two-speed BBM

Abstract: We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $\s_1$ for $s\leq bt$ and $\s_2$ when $bt\leq s\leq t$. In the case $\s_1>\s_2$, the process is the concatenation of two BBM extremal processes, as expected. In the case $\s_1<\s_2$, a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.

Richard Kraaij (TU Delft): Stationary product measures for conservative particle systems

Abstract: For the exclusion process and the independent random walk model, among others, it is known that there exist a family of stationary product measures. For the two given examples these are the Bernoulli and Poisson product measures indexed by appropriate density profiles. These models turn out to have a common structure and it turns out that every model of this type has stationary product measures. I will describe what these measures are and which density profiles are appropriate. After that, I will focus on the question whether these stationary product measures are ergodic.

Piotr Miłoś (University of Warsaw): Delocalisation of the two-dimensional Lipschitz model

Abstract: We consider a random two-dimensional surface satisfying a Lipschitz constraint. The surface is uniformly chosen from the set of all real-valued Lipschitz functions on a two-dimensional discrete torus. Our main result is that the surface delocalizes, having fluctuations whose variance is at least logarithmic in the size of the torus. The result answers an open question mentioned by Brascamp, Lieb and Lebowitz. In the talk I will also outline the proof method which follows closely the approach of Richthammer, who developed a variant of the Mermin-Wagner method applicable to hard-core constraints. I will further address potential extensions and open questions concerning tother surface models. This is a joint work with R. Peled (U of Tel Aviv).

Winny O'Kelly de Galway (Universiteit Leiden): A low temperature analysis of the boundary driven Kawasaki process

Abstract: Low temperature analysis for non-equilibrium systems is virtually nonexistent. Its reason is the absence of general principles. In a recent paper a scheme was put forward to characterize the low temperature asymptotics of continuous time jump processes under the condition of local detailed balance. I will start from that same framework to characterize the low temperature stationary condition of a one-dimensional boundary driven Kawasaki dynamics. We analyse the low temperature behaviour of the boundary driven Kawasaki process.

Tal Orenshtein (Weizman Institute and TU Munich): 0-1 Law for Directional Transience of 1-dimensional Excited Random Walks

Abstract: We will discuss the following theorem which solves a problem posed by Kosygina and Zerner. For a one dimensional excited random walk in stationary ergodic and elliptic cookie environment, the probability for transience to the right (left) is either zero or one. Joint work with Gideon Amir and Noam Berger.

Ron Peled (Tel Aviv University): Topological structure of proper 3-colorings in high dimensions

Abstract: It has recently been shown that a uniformly chosen proper 3-coloring of Z_n^d, the discrete torus, has a very rigid structure in high dimensions. Specifically, in a typical coloring just a single color will appear on almost all of the odd or even sublattice. This result was proven only for colorings restricted to satisfy certain boundary conditions. We prove that the result continues to hold when no boundary conditions are imposed. Somewhat surprisingly, the main obstruction to the proof is of a topological nature. To tackle it, we develop elements of algebraic topology in the discrete setting. The resulting discrete theory is elegant and may be of use in other settings as well. Joint work with Ohad Feldheim. No prior knowledge of proper 3-colorings or algebraic topology will be assumed.

Renato Santos (Lyon): A quenched central limit theorem for random walks in random sceneries in two dimensions

Abstract: Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media that were introduced at the end of the 70’s by Kesten-Spitzer and Borodin. They are defined as follows: let $\xi = (\xi_x)_{x \in \Z^d}$ be a random field of i.i.d. random variables (the random scenery), and $S = (S_n)_{n \in \N}$ a random walk in $\Z^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n\in \N} is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \xi_{S_1} + \cdots + \xi_{S_n}$. The law of $Z$ under the joint law of $\xi$ and $S$ is called “annealed”, and the conditional law given $\xi$ is called “quenched”. Recently, central limit theorems under the quenched law were proved for $Z$ by Nadine Guillotin-Plantard and Julien Poisat in dimensions d ≥ 3. We will discuss the extension of their results to dimension $d = 2$.

Adéla Švejda (Bonn University): Convergence of clock processes in random environments on infinite graphs

Abstract: We establish general criteria for the convergence of clock processes / time changes of random dynamics in random environments on infinite graphs. This extends the work of Bovier, Gayrard and Bovier, Gayrard, Svejda, where convergence criteria were established for finite graphs.

Martin Tassy (Brown University): Ergodic Gibbs Measures on tilings by nx1 dominos

Abstract: For the classic case of 2x1 dominos the set of gibbs measure on tilings is completely understood and depends on 2 parameters known as the horizontal and vertical slope of the measure. Using a new generalization of the well known height function, we will show why such result cannot be expected when we switch to nx1 dominos and prove that the set of EGM is actually much bigger rin this case. We will also give a detailed suggestion of what could be a description of this set.

Stephen Tate (Warwick University): Multispecies Virial Expansion

Abstract: This is joint work with my supervisor Daniel Ueltschi and our collaborators Sabine Jansen and Dmitrios Tsagkarogiannis. The notion of a cluster and virial expansion for systems where there is only one type of particle is reasonably well understood from the work of Mayer and his collaborators in the 1930s. However, what is less understood is the impact of having such expansions but with more activity and density parameters for different constituent particles. This question was answered somewhat by Fuchs (1942), who covers the case of finitely many species. He introduces the generalised radii of convergence of the virial expansion and uses techniques which foreshadow the Langrange-Good inversion (1965). The Lagrange Good inversion is used to invert the cluster expansion relationships for pressure and density, to achieve the virial expansion of pressure in terms of many density parameters. The technique is general and algebraic. We can treat all series as formal and get a relationship on the coefficients. We investigate appropriate convergence criteria for the virial expansion we achieve from this, which I present and then explain where it comes from with regards to the multivariable Lagrange-Good inversion formula. For regular virial expansions there is the interpretation of the coefficients being irreducible integrals or the integrals corresponding to 2-connected graphs. We achieve the analogue of this through the dissymmetry theorem. I will briefly explain the main purpose behind the dissymmetry theorem and how it connects with the Lagrange Inversion. I will end with some ideas on what still needs to be achieved after this work and what questions have been left over.

Christoph Temmel (VU Amsterdam): Cluster expansion via tree-operators in an abstract polymer system

Abstract: We present improved bounds of the radius of absolute and uniform convergence of the cluster expansion of an abstract polymer system around zero fugacity. This corresponds to a high temperature expansion of a hardcore (lattice) gas. The key is a better exploitation of Penrose's partition scheme of the spanning subgraph complex of a connected graph, adapated to the particular structure of the clusters in the expansion.

Tobias Wassmer (University of Vienna): Phase transition for the vacant set left by random walk on the giant component of a random graph

Abstract: We study the simple random walk on the giant component of a supercritical Erd\H{o}s-R\'enyi random graph on $n$ vertices, in particular the so-called vacant set at level $u$, the complement of the trajectory of the random walk run up to a time proportional to $u$ and $n$. We show that the component structure of the vacant set exhibits a phase transition at a critical parameter $u_{\star}$: For $uu_{\star}$ it has with high probability all components small. Moreover, we show that $u_{\star}$ coincides with the critical parameter of random interlacements on a Poisson-Galton-Watson tree, which was identified in [Tassy (2010)].

Yuan Zhang (Duke University): Phase Transition in a Meta-population Version of Schellings Model

Abstract: consider a metapopulation version of the model in which a city is divided into N neighborhoods each of which has L houses. There are NL red individuals and an equal number of blue individuals. Individuals are happy if there are  cL individuals of the opposite type in their neighborhood, and move to vacant houses at rates that depend on their state and that of their destination. Our goal is to show that if L is large then as  passes through c the system goes from a homogeneous state in which all neighborhoods have  L of each color to a segregated state in which 1/2 of the neighborhoods have 1L reds and 2L blues and 1/2 with the opposite composition. We give explicit formulas for 1 and 2 but unfortunately cannot rigirously prove our result.