## 12 September, 2013

## Title: Statistical Inference for Random Walk in Random Environment and applications to DNA unzipping

## Mikael Falconnet, Université d'Évry/CNRS

During the last two decades, new statistical data emerged from DNA unzipping experiments, in which the two
strands of a DNA molecule are pulled apart, e.g. with a constant velocity, and the signal measured, e.g. the force, while breaking successive nucleotide bond pairs is recorded. The simplified mathematical model of the DNA unzipping is a one-dimensional nearest-neighbor RWRE. The observation of the trajectory of this RWRE gives information on both biophysical properties of the molecule and its composition. In this talk, we introduce briefly the model and our settings. Then, for a ballistic transient RWRE, we estimate the parameter of the random environment by a Maximum Likelihood Estimation procedure (MLE). Using the link between RWRE and Branching Process with Immigration in a Random Environment, we show the consistency for the MLE. Joint work with F. Comets, Dasha & Oleg Loukianov, C. Matias.

## 19 Septembre, 2013

## Title: Symmetries in Trees

## Oliver Bernardi, Brandeis University

In this talk, I will present two problems related to the enumeration of trees. First, I will present a counting formula for multitype Cayley trees. This formula unifies certain recent results of Bousquet-Melou and Chapuy. It has application to the multivariate Lagrange inversion formula and to the study of the profile of random trees. Second, I will present a combinatorial proof of a counting formula for the spanning trees of the hypercube. It highlights an unexpected independence property for the directions of edges in the uniform spanning tree of the hypercube.
Our proofs for both problems takes advantage of certain symmetries of the enumerative formulas: we first prove these symmetries by simple combinatorial arguments, and then deduce the general formulas from particular cases.

## 26 September, 2013

## Title: Pathwise non-uniqueness for the SPDE's of some super-Brownian motions with immigration

## Yu-Ting Chen, CRM

Uniqueness theory in stochastic partial differential equations (SPDE’s) concerns their completeness and can induce fundamental properties of solutions such as Markov property. Nonetheless, there remain no robust methods to determine uniqueness in general SPDE’s with non-Lipschitz diffusion coefficients. The most im- portant problem, open for more than two decades, is whether pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion holds. A recent work by Mueller, Mytnik, and Perkins sheds light on this difficult problem, proving, however, that pathwise uniqueness for some closely related SPDE’s fails. In contrast to these particular SPDE’s, the SPDE’s of one-dimensional super-Brownian motions with im- migration share more properties with the SPDE of super-Brownian motion, but raise additional difficulties in settling the question of pathwise uniqueness. We resolve this question in the negative, invoking a novel approach for expansions of filtrations and other new estimates.
I will first review the SPDE of super-Brownian motion and some notions of unique- ness. I will then introduce the class of super-Brownian motions with immigration con- sidered in our work and discuss our pathwise non-uniqueness result for their SPDE’s. In the rest of the talk, I will explain some arguments of our proof.

## 24 October, 2013

## Title: Scaling limit of the abelian sandpile

## Lionel Levine, Cornell UNiversity

Which functions of two real variables can be expressed as limits of
superharmonic functions from (1/n)Z^2 to (1/n^2)Z? I'll discuss joint
work with Wesley Pegden and Charles Smart on the case of quadratic
functions, where this question has a surprising and beautiful answer:
the maximal such quadratics are classified by the circles in a certain
Apollonian circle packing of the plane. I'll also explain where the
question came from (the title is a hint!).

## October 31, 2013

## Title: Eigenvectors of the 1D Random Schrodinger Operator

## Ben Rifkind, University of Toronto

We consider a model of the one dimensional discrete random
Schrodinger operator on Z_n given by H_n = L_n + V_n, where L_n is the
discrete Laplacian and V_n is a random potential. If v_k := (V_n)_{kk}
does not depend on n, the eigenvectors are localized (Carmona et al.,
1987) and the local statistics of eigenvalues are Poisson. In order to
capture the transition between localization and delocalization
Kritchevski, Valko, and Virag (2011) analyzed the model in the case
when v_k decays like n^(-1/2) and characterized the local statistics
of eigenvalues. Building from the framework developed in that paper, I
will discuss scaling limits of the corresponding eigenvectors. They
converge (in some sense) to a simple function of Brownian motion. This
is joint work with Balint Virag.

## 7 November, 2013

## Title: Fine properties of torus cover times

## David Belius, CRM

The cover time is the first time random walk on a finite connected graph has visited every vertex of that graph. Cover times have received quite a bit of attention over the past few decades. One usually investigates the asymptotic magnitude of the cover time as the number of vertices goes to infinity. In this talk I will present a result about the fluctuations of the cover time for the torus graph.

## November 14, 2013

## Title: Jigsaw percolation

## David Sivakoff, Duke University

Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdos-Renyi graph with edge probability p, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition {V}. In some generality, for puzzle graphs with N vertices of degrees about D, this probability is close to 1 or 0 depending on whether pD(log N) is large or small. We give more detailed results for the one dimensional ring and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.

## 28 November, 2013

## Title: Smoothing Equations for Large Polya Urns

## Cécile Mailler, University of Bath

This talk will focus on large two-colour Pólya urns. From the study of the asymptotic behaviour of such an urn arises a random variable denoted by W. The underlying tree structure of the urn permits to see W as the solution in law of a fixed point equation, from which we can deduce information about its moments, or about the existence of a density. This work can be done on the discrete urn itself, or on its continuous time embedding. Though the two variables W (arisen from discrete or continuous time) are different, they are related by connexions, which often permit to translate results from one W to the other.
This work is a collaboration with Brigitte Chauvin and Nicolas Pouyanne (Versailles, France).

## 5 December, 2013, CANCELLED

## Title: TBA

## Antonio Auffinger, University of Chicago