abstracts for contributed talks

Yoshihiro Abe, Research Institute for Mathematical Sciences, Kyoto University

Title: Cover times for sequences of random walks on random graphs

Abstract: We can classify cover times for sequences of random walks on random graphs into two types: One type is the class of cover times approximated by the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type is the class of cover times approximated by the maximal hitting times. These types are characterized by the volume, effective resistances, and geometric properties of random graphs. We classify some examples, such as the supercritical Galton-Watson family trees and the incipient infinite cluster for the critical Galton-Watson family tree.

Gergely Ambrus, Renyi Institute, Budapest

Title: Longest convex chains

Abstract: A classical problem in probability is to determine the length of the longest increasing subsequence in a random permutation. Geometrically, the question can be formulated as follows: given n independent, uniform random points in the unit square, find the longest increasing chain (polygonal path through the given points) connecting two diagonally opposite corner of the square, where "length" means the number of points on the chain. The variant of the problem I am going to talk about asks for the length of the longest convex chain connecting two vertices. We determine the asymptotic expectation up to a constant factor, and derive strong concentration and limit shape results. We also prove an ergodic result as well as giving a heuristic argument for the exact asymptotics of the expectation. Some of these results are joint with Imre Barany.

Roland Bauerschmidt, UBC

Title: Finite range decomposition of Gaussian fields

Abstract: I will show how to decompose the Gaussian free field on a (weighted) graph into a sum of finite range Gaussian fields, which are smoother than the original field and have spatially localized correlations.

Jă©ră©mie Bettinelli, Universite Paris Sud

Title: Scaling limit of arbitrary genus random maps

Abstract: During this talk, we will see how some results exposed by Grégory Miermont in his course generalize to some more general classes of maps, namely arbitrary genus bipartite quadrangulations as well as quadrangulations with a boundary. In particular, we will see that these classes of maps exhibit scaling limits and we will focus on their topology.

Oriane Blondel, LPMA - Paris 7

Title: Local relaxation for FA-1f out of equilibrium

Abstract: We consider the Fredrickson and Andersen one spin facilitated model (FA1f)on Z^d. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p or q=1-p respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $\nu$ different from the stationary product p-Bernoulli measure $\mu$, which has enough zeros. We then prove local convergence to equilibrium when the vacancy density q is above a proper threshold. The convergence is exponential (d=1) or stretched exponential (d>1). Joint work with N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli.

Yu-ting Chen, UBC

Title: Sharp Benefit-to-Cost Rules for the Evolution of Cooperation on Regular Graphs

Abstract: We study two of the simple rules on finite graphs under the death-birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman, and Nowak [Nature, 441 (2006) 502-505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.

Wai Fan, University of Washington

Title: Hydrodynamic limits for a reaction diffusion system

Abstract: We will investigate an interacting particle system in which two types of Brownian particles annihilate at a certain rate when they're near the interface of their respective domains. This can model the propagation of charges in a single solar cell. We will show that the particle density is the solution of a coupled PDE and obtain a probabilistic representation of the solution.

Hilary Finucane, Weizmann Institute of Science

Title: Algebraic recurrence of random walks on groups

Abstract: Consider a symmetric random walk on a group G. If the trace of the random walk generates G as a semigroup almost surely, then we say that G is algebraically recurrent. In this talk, we will present some initial steps towards understanding algebraic recurrence, including examples of algebraically recurrent and non-algebraically recurrent groups. We will conclude with some open questions. This is joint work with Itai Benjamini and Romain Tessera.

Matthew Folz, UBC

Title: Volume growth and random walks on graphs.

Abstract: We discuss various behaviours of continuous time simple random walks which are governed by the volume growth of the underlying weighted graph. In this setting the volume growth is computed with respect to a metric adapted to the random walk and not the graph metric. Use of these metrics allows us to establish results for graphs which are analogous to those for diffusions on a manifold or the Markov process associated with a strongly local Dirichlet form.

Felipe Garcia-ramos, UBC

Title: Cesaro limit and ergodicity of locally eventually periodic measures

Abstract: We study translation invariant deterministic dynamics (phi) on the lattice (cellular automata). In particular the evolution and limit of probability measures that give the set of locally eventually phi-periodic points full measure. We prove the convergence of the mean averages under phi of this measures. We characterize the ergodicity of the limit measures (solving a question posed by Blanchard and Tisseur) and we prove that in the limit phi is a mixture measure theoretical odometers.

Adriăˇn Gonzăˇlez Casanova, Berlin Mathematical School

Title: Seed bank models with long range dependence.

Abstract: In this talk I present a new model for seed banks, where individuals may obtain their type from ancestors which have lived in the near as well as the very far past. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are particular cases of the model. The mathematical methods are based not only on Markov chains, but also on renewal theory as well as on a Gibbsian approach introduced by Hammond and Sheffield (2011) in a different context. This talk is based in a joint work with Jochen Blath, Noemi Kurt, Dario Span`o.

Title: Fluctuations study for type-dependent stochastic spin models

Abstract: We study the fluctuations process for the type-dependent stochastic spin models proposed by Fernández et al.[2], which were used to model biological signaling networks. Using the results of Ethier & Kurtz [1], we analyse the asymmetric basic clock [3], a extension for the simplest cyclic-interaction module, that provides the basic functionality of generating oscillations. Particularly, we apply the central limit theorem for fluctuations process; the dynamics of this limit process is our aim. References. [1] S.N. Ethier, T.G. Kurtz, Markov Processes, Characterization and Convergence. Wiley, New York, 1986. [2] R. Fernandez, L.R. Fontes, E.J. Neves, Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories. J Stat Phys (2009) 136: 875-901. [3] M.A. González Navarrete, Sistemas de partículas interagentes dependentes de tipo e aplicaçoes ao estudo de redes de sinalizaçao biológica. Master thesis, Instituto de Matemática e Estatística USP, 2011.

Ori Gurel-gurevich, UBC

Title: Recurrence of planar graph limits

Abstract: We will show that any distributional limit of finite planar graphs in which the degree of the root decays exponentially is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation/quadrangulation (UIPT/UIPQ) are almost surely recurrent, resolving conjectures of Angel, Benjamini and Schramm. Joint work with Asaf Nachmias.

Xin He, Auburn University

Title: Lebesgue approximation of $(2,\beta)$-superprocesses

Abstract: Let $\xi=(\xi_t)$ be a locally finite $(2,\beta)$-superprocess in $\RR^d$ with $\beta<1$ and $d>2/\beta$. Then for any fixed $t>0$, the random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of ${\rm supp}\,\xi_t$. This extends the Lebesgue approximation of Dawson-Watanabe superprocesses. Our proof is based on a truncation of $(\alpha,\beta)$-superprocesses and uses bounds and asymptotics of hitting probabilities.

Tyler Helmuth, UBC

Title: Ising Models via Heaps of Pieces

Abstract: In the 60s Sherman gave a purely combinatorial method for solving the Ising model on planar graphs using non-backtracking walks, and recently Cimasoni and Loebl have described similar methods for graphs embedded in surfaces. I'll outline an alternative proof of these results using the theory of heaps of pieces, and will show how the result gives expressions for spin correlation functions in terms of non-backtracking walks.

Jacob Kagan, The Weizmann Institute of Science

Title: Monotonic IDLA forest and First Passage Percolation

Abstract: We present a modification of the IDLA model on a rotated square lattice. The process results in a forest of trees covering the upper half plane. We show an equivalence between this model and first passage percolation. We prove that with probability 1 the trees resulting from the IDLA forest are finite.

Igor Kortchemski, Université Paris-Sud (Orsay, France)

Title: Limit theorems for conditioned non-generic Galton-Watson trees

Abstract: We are interested in a particular type of subcritical Galton-Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. We investigate this phenomenon by studying scaling limits of such trees. In particular, we show that the height of such trees grows logarithmically in their size.

Dorota Kowalska, Warsaw University of Technology

Title: Density of states for alpha-stable processes

Abstract: We will show the existence of the density of states for $alpha$-stable processes ie existence of the deterministic measure that is a limit (when M goes to infinity) of random measures based on sequence of the eigenvalues of the generator of the $alpha$-stable process in a ball B(0,M) with Poissonian obstacles. We will give also estimate of the limit measure near zero.

Marcin Lis, VU University Amsterdam

Title: Correlation functions in the 2D Ising model via signed loops and paths.

Abstract: Using the combinatorial method for the 2D Ising model originating in the works of Sherman, Burgoyne and others we derive formulas for the correlation functions in terms of signed loops and paths. In the case of regular lattices we also identify the critical temperature for the phase transition in the long range behavior of these functions. Joint work with Wouter Kager and Ronald Meester.

Sergio Lopez, Universidad Nacional Autonoma de Mexico (UNAM)

Title: Distributional fixed points and attractors in queueing theory.

Abstract: (Joint work with Pablo Ferrari) In 1956, Burke proved that the departure process of a stationary queue with Poisson arrivals is a Poisson process. In other words, Poisson process is a distributional fixed point for ./M/1 operation. Mountford and Prabhakar's theorem ('95) showed that Poisson process is not only a fixed point but an attractive distribution over some wide class of ergodic point processes on the line. Some extensions to non-markovian servers were made by Mairesse and Prabhakar ('03) for the existence of fixed points, and Prabhakar ('03) for the attractiveness. In 2001 O'Connell and Yor showed some brownian analogues to Burke's theorem. For a Brownian queue, Brownian motion is a distributional fixed point. In this talk we will show some results about the attractiveness of Brownian motion under the Brownian queue operation.

Shuwen Lou, University of Washington

Title: Multi-dimensional Brownian Motion with Darning

Abstract: The reason that we define multi-dimensional Brownian motion as a darning process is that, even for the simplest case which is R^2 being unioned with R^1, such a process cannot be defined in the usual sense, because 2-dimensional Brownian motion never hits a singleton. Constructions of darning processes are based on one-point extension theory which was first studied by M. Fukushima. Lots of very interesting examples, for instance, circular Brownian motion, Brownian motion with a ``knot", etc., can be constructed in this way, some of which will be provided in the talk. The rest of the talk will be focusing on the heat kernel estimates of multi-dimensional Brownian motion with darning.

Yuri Mejia, UBC

Title: The critical points of lattice trees and lattice animals in high dimensions

Abstract: Lattice trees and lattice animals are used to model branched polymers. They are of interest in combinatorics and in the study of critical phenomena in statistical mechanics. A lattice animal is a connected subgraph of the d dimensional integer lattice. Lattice trees are lattice animals without cycles. We consider the number of lattice trees and animals with n bonds that contain the origin and form the corresponding generating functions. We are mainly interested in the radii of convergence of these functions, which are the critical points. In this talk we focus on the calculation of the first three terms of the critical points for both models as the dimension goes to infinity. This is ongoing work with Gordon Slade.

Raoul Normand, Univ. Paris VI & Univ. of Toronto

Title: A model of migration under constraint

Abstract: We will present a random model of population, where individuals live on several islands, and will move from one to another when they run out of resources. Our main goal is to study how the population spreads on the different islands, when the number of initial individuals and available resources tend to infinity. Finding this limit relies on asymptotics for critical random walks and (not so classical) functionals of the Brownian excursion.

Miklos Racz, UC Berkeley

Title: Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction

Abstract: We introduce and investigate a model of a finite number of competing particles jumping forward on the real line. The evolution of the particles is a continuous-time Markov jump process: given a configuration, each particle jumps with a rate that depends on the particle's relative position compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. Whenever a jump occurs, the jump length is chosen independently of everything else from a positive distribution. Real-life phenomena that could be modeled this way includes the evolution of prices in a market, or herding behavior of animals. The main point of interest is the behavior of the model as the number of particles goes to infinity. We prove that in this fluid limit the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. We also present a surprising connection to extreme value statistics. This is joint work with Marton Balazs and Balint Toth.

Gourab Ray, University of British Columbia

Title: Half Planar Triangulations

Abstract: Let $\boldsymbol{T_{m,n}}$ be an uniformly picked triangulation of an $m-$gon with $n$ internal vertices rooted on the boundary. Suppose $\{m,n\}$ converge to $\{\infty, \infty\}$ with $m/n$ converging to some number $a \in [0,\infty]$. We show that the local limit of $\boldsymbol{T_{m,n}}$ exist and the limit only depends on $a$ but not on the sequence. For $a=0$ the limit coincides with that of half planar UIPT. For $a>0$ we show that the boundary of the balls in graph distance around the root form a tight sequence and also show that they grow roughly like square of the radius. We also obtain a distributional limit of the properly centered and scaled ball volumes. We show that these maps satisfy two special properties namely translation invariance and domain Markov property. We show that any measure supported on half planar triangulations satisfying translation invariance and domain Markov property must either be one of the maps obtained above or is a new class of triangulations which we name as h-triangulations. We also show that h-triangulations have exponential volume growth giving it a hyperbolic structure.

Daisuke Shiraishi, Kyoto university

Title: Cut points for simple random walks

Abstract: We consider two random walks conditioned “never to intersect” in Z^2. We show that each of them has infinitely many `global' cut times with probability one. In fact, we prove that the number of global cut times up to n grows like n^{3/8}. Next we consider the union of their trajectories to be a random subgraph of Z^2 and show the subdiffusivity of the simple random walk on this graph.

Debleena Thacker, Indian Statistical Institute, Delhi.

Title: On PĂłlya Urn Schemes with Infinitely Many Colors.

Abstract: In this talk, we extend the mutlicolor P/'olya urn schemes to countably infinitely many colors. We index the colors by \mathbb{Z}. Throughout the talk, we discuss mainly replacement matrices arising out of random walks. We show that the proportion of colors with suitable centering and scaling show central tendencies. Also the centering and scaling are fairly general. This behavior is in sharp contrast with the finite color case, where the asypmtotic behavior of the proportion of colors are determined by the qualitative properties (transience or recurrence) of the Markov chain underlying the replacement matrix. We also extend the infinite color case to fairly general graphs on \mathbb{R}^{d} and show that the proportion of colors show central tendencies similar to that in the case for \mathbb{Z}. Even the centering and scaling remains same.

Daniel Valesin, UBC

Title: Extinction time of the contact process on finite trees

Abstract: We consider the contact process on finite trees. We assume that the infection rate is larger than the critical rate for the one-dimensional process. We show that, for any sequence of trees with increasing number of vertices and degree bounded by a universal constant, the expected extinction time of the process grows exponentially. Additionally, the extinction time divided by its expectation converges in distribution to the unitary exponential distribution. This is joint work with Thomas Mountford, Jean-Christophe Mourrat and Qiang Yao.

Tim Van De Brug, VU University Amsterdam

Title: 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. Joint work with Erik Broman, Federico Camia, Matthijs Joosten and Ronald Meester.

Xu Yang, Beijing Normal University

Title: Stochastic Equations of Super-L\'{e}vy Process with General Branching Mechanism

Abstract: The process of distribution functions of a one-dimensional super-L\'{e}vy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space white noises and Poisson random measures. This generalizes a recent result of Xiong (2012), where the result for a super-Brownian motion with binary branching mechanism was obtained. To establish the main result, we prove a generalized It\^o's formula for backward stochastic integrals and study the pathwise uniqueness for a general backward doubly stochastic equation with jumps. Furthermore, we also present some results on the SPDE driven by a one-sided stable noise without negative jumps. This is a joint work with Hui He and Zenghu Li.