September 7, 2012
Joint with Séminaire Analysis (McGill & Concordia)
Localisation in the parabolic Anderson model
Nadia Sidorova, University College, London
The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the mass transport through a random field of sinks and sources and is actively studied in mathematical physics. We discuss, for a class of i.i.d. potentials, the intermittency effect occurring for such potentials, which manifests itself in increasing localisation and randomisation of the solution.
October 4, 2012
The evolution of genealogical trees under selection
Peter Pfaffelhuber, Freiburg
We construct a tree-valued Markov process describing the
evolution of genealogical relationships in populations of constant
size. While the case of equally fit individuals is termed neutral
evolution, we concentrate on the case where the distribution in
offspring numbers depend on the type of an individual. In the talk we
will develop similarities to the measure-valued Fleming-Viot process
with mutation and selection, which will also be explained in
detail. This is joint work with Andrej Depperschmidt and Andreas
October 18, 2012
Subdiffusivity of random walk on the 2D invasion percolation cluster
Phil Sosoe, Princeton University
In the 1980s, H. Kesten showed that the simple random walk on the incipient infinite cluster, an infinite random graph closely related to critical percolation, is subdiffusive in an averaged sense. With Jack Hanson and Michael Damron from Princeton University, we have extended this result to another critical model, invasion percolation. I will discuss this result, as well as various improvements of Kesten's method we have discovered, including a simple proof of an almost sure version of Kesten's subdiffusivity theorem.
CANCELLED: November 1, 2012
Voter model perturbations in one-dimension
Emmanuel Schertzer, Princeton University Paris VI
In a recent work, Cox, Durrett and Perkins consider a class of interacting particle systems in dimension 3 and higher, and whose rate of switching can be written as a perturbation of the classical linear voter model. For such systems, the authors show that the (rescaled) local density of 1's converges to the solution of an explicit reaction diffusion equation.
In this talk, we discuss the case d=1 and show that voter model perturbations can not be described in terms of a reaction diffusion equation anymore, but directly in terms of a duality relation with the Brownian net : an infinite family of one-dimensional coalescing-branching Brownian motions. Several applications to Ecology and Physics will be discussed.
This is joint work with C. Newman and K.Ravishankar.
November 8, 2012
The uniform spanning tree in two dimensions
Martin Barlow, University of British Columbia
This talk will discuss properties of the UST on the
Euclidean lattice, and in particular with the relation
between distance in the tree and Euclidean distance.
These results can then be applied to study SRW on the UST.
Joint work with Robert Masson
November 15, 2012
Large and Moderate Deviations for Some Measure-Valued
Parisa Fatheddin, University of Tennessee
Large and Moderate Deviations are considered in this talk for a class of stochastic
differential equations (SPDE) having non-Lipschitz continuous coefficients. Since
two important population models, super-Brownian Motion and Fleming-Viot process,
can be represented by this class of SPDEs, then as an application, we derive Large
and Moderate Deviations for these models.
November 22, 2012
Gaussian free field, random measure and KPZ on R^4
Linan Chen, McGill University
A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics model in Euclidean geometry and its counterpart in the random geometry. Recently, Duplantier and Sheffield used the 2D Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. We have applied a similar approach to generalize part of their results to R^4 (as well as to R^(2n) for n>=2). To be specific, we construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) given by the exponential of an instance of the 4D Gaussian free field. We also establish the KPZ relation corresponding to this random measure. This is joint work with Dmitry Jakobson.
November 29, 2012
From Quicksort to real trees
Henning Sulzbach, McGill University
The contraction method for sequences of random variables satisfying additive distributional recurrences was introduced by Rösler (1991) in the probabilistic analysis of the well-known Quicksort algorithm. In this talk, we give a short overview of the method in finite dimensional state spaces including various applications with different origins. We then focus on recent generalizations of the framework to recurrences on a functional level relying on a family of probability metrics introduced by Zolotarev in the late seventies. We finally present a recent development concerning the scaling limit of random trees in the context of recursive partitionings of the the disk as introduced by Curien and Le Gall (2011).