SCIENTIFIC PROGRAMS AND ACTIVITIES

Balint Virag
Covariance structures for iid factors on regular trees

There is a one-parameter family of 0-1 valued Markov processes indexed by the d-regular tree. We would like to construct these processes as an invariant, deterministic function (factor) of iid random variables on the vertices. It is an open problem to determine when this is possible.

I will review how this is connected to independent sets, max and min cut problems on large girth graphs, and Benjanmini-Schramm convergence. While it is still mystery what processes can be a factor of iid, a lot more is known about when a covariance structure has this property. Join work with Agnes Backhausz and Balazs Szegedy.

January 13

Lerna Pehlivan (Washington)
Structure of Random 312-Avoiding Permutations

A permutation of {1,2,..,N} is said to avoid 312 pattern if there is no subsequence of three elements of this permutation that appears at the same relative order as 312. Monte Carlo experiments reveal some features of random 312 avoiding permutations. In light of these experiments we determine some probabilities explicitly.

This paper is a joint work with Neal Madras.

Andrew Stewart (Toronto)
The scaling limit of the range of the simple random walk bridge
on regular trees

We consider the lazy simple random walk bridge of length n on a d-regular tree. The range of the simple random walk bridge, the set of vertices $R_n$ visited by a bridge of length $n$, is a finite tree whose diameter is $\approx \sqrt{n}$. We show that the metric space $R_n/\sqrt{n}$ converges in distribution in the Gromov-Hausdorf metric to the Brownian Continuum Random Tree introduced by Aldous. We use techniques introduced in [1].
This is joint work with Balint Virag.

[1] Aldous, D. The continuum random tree III. Ann. Probab. Volume 21,
Number 1 (1993), 248-289.

We discuss spectral properties of some block matrices whose entries are random Schroedinger operators. These operators model certain systems connected to superconductor physics. We will concentrate on matrices in the BCS form which arises in the theory of
superconductors. We concentrate on properties of the density of states for those operators as well as on the question of Anderson localization.

Tom Bloom (Toronto)
Large deviation for outlying coordinates in ß ensembles

Abstract: ß ensembles are generalizations of the joint probability distribution of the eigenvalues of the GOE and GUE. A ß ensemble on a compact set K?C is a probability distribution Prob_{n,ß} on K_n for n=1,2...
We show that the related sequence of probability distributions on K defined by, for W $?$ K, Prob_{n,ß}(z1$ ? $W) where z1 is the first coordinate, satisfies a large deviation principle with speed n and an explicit rate function. This extends work of Borot-Guionnet from ß ensembles on R.

Almut Burchard (Toronto)
Some applications of two-point symmetrization in Probability

Two-point symmetrization is a simple equimeasurable rearrangement of sets and functions that pushes mass towards the origin. It is often used to prove that certain symmetric functionals have radially symmetric extremals; it can be particularly helpful for identifying equality cases. I will describe a rearrangement inequality for multiple integrals, and discuss classical and recent applications to geometric problems involving path integrals.

Ben Rifkind
Eigenvectors of the 1D Random Schrodinger Operator

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.

Van Vu
How many real roots does a random polynomial have?

Consider a polynomial P_n = c_0 + c1x +...c_n x^n of degree n whose coefficients c_i are (not necessarily iid) real random variables. The problem of determining N_n, the number of real zeroesof P_n goes back to Waring (1782), and has become popular since the series of works of Littlewood and Offord in the 1940s. Deep works of Littlewood-Offord, Erdos-Offord, Turan, Kac, Stevens, Ibragimov-Maslova, Edelman-Kostlan and many others give us a good understanding of N_n in the case c_i are iid random variables with mean 0 and variance 1 (see John Baez's "Lord of the Ring" beautiful picture on http://math.ucr.edu/home/baez/). However, much less is known for the all other cases, when the c_i may have different variances and/or are dependent (a good example is the characteristic polynomial of a random matrix).

In this talk, I am going to give a brief survey on the state of the art of the problem, and introduce a new approach, developed together with T. Tao, that leads to a very good understanding of real (and also complex) roots of a very general class of random polynomials.

An independent set in a graph is a set of vertices such that there are no edges between them. How large can an independent set be in a random d-regular graph? How large can it be if we are to construct it using a (possibly randomized) algorithm that is local in nature? The talk will disuss a recently introduced notion of local algorithms for combinatorial optimization problems on large, random d-regular graphs. The talk will then explain why, for asymptotically large d, local algorithms can only produce independent sets of size at most half of the largest ones. The factor of 1/2 turns out to be optimal. Joint work with Balint Virag.

The Khardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation used to describe randomly evolving interfaces. Its solution has an unusual scaling behaviour, and the distribution of the fluctuations are related to random matrices. The class of such models is called the KPZ universality class and is predicted to contain a number of discrete and semi-discrete models. We will discuss some of these models and recent progress made towards establishing this universality for the so-called half-flat and flat initial conditions for the Asymmetric Simple Exclusion Process, based on joint work with Jeremy Quastel and Daniel Remenik.