Friday September 4, 2015 
Peter Forrester (Melbourne) 
Raney distribution and random matrices

The Raney numbers are a generalisation of the FussCatalan numbers,
which occur in ballot type problems. Recently they have been shown to occur in random matrix
theory as the moments of eigenvalue probability densities. I'll develop some themes that
come out of this interpretation.

Friday September 11, 2015 
Mark Freidlin (Maryland) 
Longtime influence of small perturbations

I will consider deterministic and stochastic perturbations of
dynamical systems and stochastic processes. In many models, longtime
evolution of
the perturbed system has two components: a slow one which is a motion
on the simplex of invariant measures of the nonperturbed system, and
the
fast one which can be characterized by the invariant measure related
to the position of slow component. Under certain assumptions, the slow
component, in an appropriate time scale, converges to a Markov process
on the simplex of invariant measures.
Transitions between asymptotically stable regimes which are due to
large deviations, exit problems, classical and recent results on the
averaging
principle, homogenization problems, interface motion in reaction
diffusion equations, and other models can be considered in this
framework.
I will consider some of these problems in more details.

Wednesday September 16, 2015 
Laurent Menard (Paris) 
Percolation by cumulative merging and phase transition for the contact process on random graphs

We establish a connection between the contact process on any (locally finite) graph and a new percolation model. We call it cumulative merging as clusters are built by recursive merging of smaller sets of vertices. First, we will see that large clusters describe the geometry of the sets where the contact process survives for a long time. Then, we will show that this percolation model has a non trivial phase transition for several classical (deterministic and random) graphs.
As an application, we prove that the contact process has a non trivial phase transition on ddimensional Delauney triangulations and geometric graphs. To the best of our knowledge, these are the first examples of graphs with unbounded degrees where this is the case.
This is joint work with Arvind Singh (Orsay).

Friday September 18, 2015 
10 AM: Rongfeng Sun (Singapore) 
Scaling Limits of Disordered Systems: Marginal Relevance and Universality

We consider disordered systems of directed polymer type, for which disorder is socalled marginally relevant. This includes the disordered pinning model with tail exponent 1/2, the usual (shortrange) directed polymer model in dimension two, and the longrange directed polymer model with Cauchy tails in dimension one. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a lognormal random field with a multiscale correlation structure, which undergoes a phase transition as the disorder strength varies. As a byproduct, we show that the solution of the twodimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. Joint work with F. Caravenna and N. Zygouras.


11 AM: Arno Kuijlaars (Leuven) 
Products of random matrices

I will present an overview of recent work on singular values of
products of complex Ginibre random matrices. This development
started with the work of Akemann, Ipsen, Kieburg and Wei in 2013
who showed that the squared singular values are a determinantal
point process with correlation kernel that can be expressed
in terms of Meijer Gfunctions.
The correlation kernels have explicitly computable scaling limits
as the sizes of the matrices tend to infinity. One finds the
familiar sine kernel in the bulk and the Airy kernel at the soft edge
of the limiting spectrum. At 0, which is a hard edge, there is a new
family of limiting that are again expressible in terms of Meijer Gfunction.
These Meijer Gkernels are univeral and they also appear in biorthogonal
ensembles and in multimatrix models.

Friday September 25, 2015 
Wouter Kager (Amsterdam) 
Signed loop representations of Ising correlations

The Ising model is one of the simplest models in statistical physics
that exhibits a phase transition. In this talk, we will focus on the
spinspin correlations in the model, and derive a representation for
these correlations in terms of signed loops on the underlying graph.
This representation leads to an appealing proof of the fact that the
phase transition in the magnetic behavior is sharp, that works for a
surprisingly wide class of graphs.
The origins of the signed loop approach to the Ising model can be traced
back to a paper by Kac and Ward from 1952. The involved combinatorics is
rather complicated, however, and the method was abondoned soon after the
discovery of the dimer approach. Interest in the method has been revived
in recent years in the probability community. The work presented in this
talk is joint work with Marcin Lis and Ronald Meester.

Friday October 2, 2015 
Courant Instructor Day 

Friday October 9, 2015 
ColumbiaCourant probability seminar series 

Friday October 16, 2015 
Roland Bauerschmidt (Harvard) 
Local eigenvalue statistics for random regular graphs

I will discuss results on local eigenvalue statistics for uniform
random regular graphs. Under mild growth assumptions on the degree,
we prove that the local semicircle law holds at the optimal scale,
and that the bulk eigenvalue statistics (gap statistics and averaged
energy correlation functions) are given by those of the GOE. This is
joint work with J. Huang, A. Knowles, and H.T. Yau.

Friday October 23, 2015 
Ivan Nourdin (Luxembourg) 
The Gaussian Product Conjecture

Let Z=(Z_1,…,Z_d) be any centered Gaussian vector and let m be an integer bigger or equal to 1. The Gaussian Product Conjecture (GPC), which is still open, claims that the 2m moment of the product Z_1 … Z_d is always bigger than the product of the 2m moments. In this talk, we will first explain why this conjecture implies another conjecture from Banach space theory, the socalled Real Polarization Conjecture. Then we will prove an inequality related to GPC and involving Hermite polynomials, and we will review several consequences. Based on a joint work with Dominique Malicet (Rio de Janeiro), Giovanni Peccati (Luxembourg) and Guillaume Poly (Rennes).

Friday October 30, 2015 
Nicholas Cook (UCLA) 
Random regular digraphs: singularity and spectrum

We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph is dense. Towards establishing the same result for the unweighted adjacency matrix, we prove that it is invertible with high probability, even for sparse digraphs with degree growing only polylogarithmically. 
Friday November 6, 2015 
Guillaume Barraquand (Columbia) 
Random walks in Beta random environment

We consider a model of random walks in spacetime random environment, with Betadistributed transition probabilities. This model is exactly solvable, in the sense that the law of the (finite time) position of the walker can be completely characterized by Fredholm determinantal formulas. This enables to prove a limit theorem towards the TracyWidom distribution for the second order corrections to the large deviation principle satisfied by the walker, thus extending the scope of KPZ universality to RWRE. We will also discuss a few similar results about degenerations of the model: a first passage percolation model which is the "zerotemperature" limit, and a certain diffusive limit which leads to wellstudied stochastic flows. (Work in collaboration with Ivan Corwin) 
Friday November 13, 2015 
10 AM: Yuri Kifer (Jerusalem) 
Further advances in nonconventional limit theorems


11 AM: Shirshendu Ganguly (Seattle) 
Competitive erosion is conformally invariant

We study a graphtheoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite, competitive version of the celebrated Internal DLA growth model first analyzed by Lawler, Bramson and Griffeath in 1992.
We establish conformal invariance of competitive erosion on discretizations of smooth, simply connected planar domains. This is done by showing that at stationarity, with high probability the red and the blue regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. The proof relies on convergence of solutions of the discrete Poisson problem with Neumann boundary conditions to their continuous counterparts and robust IDLA estimates. (Joint work with Yuval Peres, available at http://arxiv.org/abs/1503.06989).

Friday November 20, 2015 
Northeast probability seminar 

Friday November 27, 2015 
Thanksgiving 

Friday December 4, 2015 
10 AM: Oren Louidor (Technion) 
Aging in a logarithmically correlated potential

We consider a continuous time random walk on the box of side length N in Z^2, whose transition rates are governed by the discrete Gaussian free field h on the box with zero boundary conditions, acting as potential: At inverse temperature \beta, when at site x the walk waits an exponential time with mean \exp(\beta h_x) and then jumps to one of its neighbors chosen uniformly at random. This process can be used to model a diffusive particle in a random potential with logarithmic correlations or alternatively as Glauber dynamics for a spinglass system with logarithmically correlated energy levels. We show that at any subcritical temperature and at preequilibrium time scales, the walk exhibits aging. More precisely, for any \theta > 0 and suitable sequence of times (t_N), the probability that the walk at time t_N(1+\theta) is within O(1) of where it was at time t_N tends to a nontrivial constant as N \to \infty, whose value can be expressed in terms of the distribution function of the generalized arcsine law. This puts this process in the same aging universality class as many other spinglass models, e.g. the random energy model. Joint work with Aser CortinesPeixoto and Adela Svejda.


11 AM: Robert Neel (Lehigh University) 
Random walks, Laplacians, and volumes in subRiemannian geometry

We study a variety of random walks on subRiemannian manifolds and their diffusion limits, which give, via their infinitesimal generators, secondorder operators on the manifolds. A primary motivation is the lack of a canonical Laplacian in subRiemannian geometry, and thus we are particularly interested in the relationship between the limiting operators, the geodesic structure, and operators which can be obtained as divergences with respect to various choices of volume. On the stochastic side, we give a fairly general convergence scheme for limits of random walks using martingale methods, and we also see some intuitivelyappealing "discrete versions" of Ito and Stratonovich SDEs and of Girsanov's theorem.

Friday December 11, 2015 
10 AM: Zhipeng Liu (Courant) 
Asymptotics in periodic TASEP with step initial condition

We consider the periodic TASEP model on the space \{(x_1,\cdots,x_N)\in Z^N; x_1<\cdots<x_N<x_1+M\}. This model can be described as the TASEP on the ring with length M, or directed last passage percolation with periodic entries, or directed last passage percolation on a cylinder. We are interested in the fluctuations of a fixed particle as N, M, and time t all go to infinity. For the step initial condition, if the density of particles N/M is fixed, we find the explicit formula for this limiting distribution in the critical case t=O(N^{3/2}). This is a joint work with Jinho Baik.


11 AM: Brian Rider (Temple) 
A universality result for the random matrix hard edge

The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and "quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by José Ramírez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.
