By Shige PENG, Shandong University
By Rongfeng SUN, National University of Singapore
The lonely branching random walks on Z^d is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. It has been conjectured that in dimensions 1 and 2, the lonely branching random walks die out locally. We prove this conjecture and show furthermore that the same result holds if additional branching is allowed when the walk is not alone.
Joint work with Matthias Birkner.
By Emmanuel Schertzer, Université Paris VI
Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this talk, I will focus on the height and contour processes encoding a general CMJ forest.
I will first show that the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. I will present two applications of this result: (1) in the case of ``short'' edges, the height process of a CMJ is obtained by stretching by a constant factor the height process of the associated genealogical Galton–Watson tree, and (2) when the offspring distribution has a finite second moment, the genealogy of the CMJ can be obtained from the underlying genealogical structure by a marking procedure, related to the so-called Poisson snake.
This is joint work with Florian Simatos.
First Passage Percolation on Exponential of Log-correlated Gaussian Fields in Two-dimensional Lattice
By Jian DING, The University of Chicago
I will discuss first passage percolation problems in two-dimensional lattice where the vertex weight is given by exponentiating log-correlated Gaussian fields. I will present some recent progress on the exponent for the weight of the geodesic including an upper bound as well
as a non-universality result.
This is based on joint works with Subhajit Goswami and Fuxi Zhang.
By Michael Damron, Georgia Institute of Technology
In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the `90s, Licea and Newman showed that, under a curvature assumption on the ``asymptotic shape,'' all infinite geodesics have asymptotic directions and there are no bigeodesics with both ends directed in some deterministic subset D of [0,2pi) with countable complement. I will discuss recent work with Jack Hanson in which we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction.
By Elisabetta Candellero, University of Warwick
In this talk we will discuss some relations between percolation on a given graph G and its geometry.
There are several interesting questions relating various properties of G such as growth (or dimension) and the process of percolation on it.
In particular we will look for conditions under which its critical percolation threshold is non-trivial, that is: p_c(G) is strictly between zero and one.
In a very influential paper on this subject, Benjamini and Schramm asked whether it was true that for every graph satisfying dim(G) > 1, one has p_c(G) < 1. We will explain this question in detail and present some recent results that have been obtained in this direction.
This talk is based on a joint work with Augusto Teixeira (IMPA, Rio de Janeiro, Brazil).
By Alexandre Stauffer, University of Bath
We consider a stochastic aggregation model on Z^d.
Start with an infinite collection of particles located at the vertices of the lattice, with at most one particle per vertex, and initially distributed according to the product Bernoulli measure with parameter $\mu\in(0,1)$.
In addition, there is an aggregate, which initially consists of only one particle placed at the origin.
Non-aggregated particles move as continuous time simple symmetric random walks obeying the exclusion rule, whereas aggregated particles do not move.
The aggregate grows indefinitely by attaching particles to its surface whenever a particle attempts to jump onto it.
Our main result states that if on Z^d, d at least 2, the initial density of particles \mu is large enough, then with positive probability the aggregate grows with positive speed.
This is a joint work with Vladas Sidoravicius.