Special Session on Random Processes: AMS Sectional: Holy Cross, April 9-10, 2011

2011 Spring Eastern Sectional Meeting:  Worcester, MA, April 9-10, 2011 (Saturday - Sunday); Meeting #1070
Special Session on Random Processes (Code: SS 7A)
Organizers: Andrew Ledoan, Boston College; Steven J. Miller, Williams College; Mihai Stoiciu, Williams College

Abstract Submissions:  http://www.ams.org/meetings/sectional/2177_deadlines.html

AMS Conference Homepage: http://www.ams.org/meetings/sectional/2177_program.html

Info on Housing / Registration / Other: http://www.ams.org/meetings/sectional/2177_other.html

Program for our session: http://www.ams.org/meetings/sectional/2177_program_ss7.html#title

Speakers for our Session (titles and abstracts):

• Alex Bloemendal (Toronto): Finite rank perturbations of large random matrices: Finite (or fixed) rank perturbations of large random matrices arise in a number of applications. The main phenomenon is a phase transition in the largest eigenvalues as a function of the strength of the perturbation. I will describe joint work with Balint Virag in which we introduce a new way to study these models. The starting point is a reduction to a natural band form; under the soft edge scaling, it converges to a souped-up version of the known continuum random Schodinger operator on the half-line. We describe the near-critical uctuations in several ways, solving a known open problem in the real case. One characterization -- a simple linear PDE -- also yields a new route to the Painleve structure in the celebrated Tracy-Widom laws.
• Paul Bourgade (Harvard): Extreme spacings for eigenvalues of random matrices: This is a joint work with Gerard Ben Arous about the extreme gaps between eigenvalues of random
  matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the smallest gaps, when rescaled by n^-4/3 , are Poissonian with an explicit limiting density. Concerning the largest gaps, normalized by sqrt(n)/n, they converge in L^p to a constant for all positive p. We compare these results with the extreme gaps between zeros of the Riemann zeta function.
• Wlodek Bryc (Cincinnati): Meixner matrix ensembles: In this talk I will discuss random matrices that are matricial analogs of the well known binomial, Poisson, and negative binomial
  random variables. The defining property is that the conditional variance of X given the sum S=X+X' of two independent copies of X is a quadratic polynomial in S; this property describes the family of six univariate laws on R that will be described in the talk, and we are interested in their matrix analogs. The talk is based on joint work with Gerard Letac.
• Sunil Chhita (Brown): Particle Systems arising from an Anti-ferromagnetic Ising Model: We present a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model has a bijection with a one-dimensional particle system equipped with creations and annihilations. We give the exact phase diagram, which determines two signicant values - the independent and critical values. We also present some results for the behavior of the model in the scaling window.
• Adrien Kassel (ENS): Geometrical properties of certain determinantal processes: We introduce qdeterminantal processes, which are a generalization of determinantal and Pfaffian processes, where the usually complex-valued kernel now takes quaternionic values. Our purpose is to give a better description of certain random point processes living on embedded graphs. In this light, we present results on two models: the CRSF (a geometrical generalization of the uniform spanning tree introduced recently by Richard Kenyon), and dimers on bipartite graphs. We also comment on applications to other Pfaffian processes and address the question of sampling.
• Richard Kenyon (Brown): Cycle-rooted spanning forests: The cycle-rooted spanning forest (CRSF) on a graph is a natural generalization of a spanning tree. It is a subset of the edges of a graph in which each component has as many vertices as edges. We study natural determinantal probability measures on CRSFs. We show that in appropriate scaling limits on the square grid, the cycle structure is conformally invariant.
• Murat Kologlu and Steven J. Miller (Williams): Distributions of Eigenvalues of Real Symmetric m-Circulant Matrices: Random matrix ensembles model many phenomena, from nuclear energy levels to L-function zeros. The idea is to generate N x N matrices from some nice distribution and look at their spectra. As N tends to infinity, the behavior of the eigenvalues of a typical matrix is close to the ensemble average. However, few ensembles are well-understood, and current theorems rarely illustrate transitions between ensembles. We study real symmetric $m$-circulant matrices with entries i.i.d.r.v. An m-circulant matrix has toroidal diagonals periodic of period m. We view m as a dial we can turn from the highly structured real symmetric circulant matrices to the ensemble of all real symmetric matrices. The limiting eigenvalue densities p_m show a visually stunning convergence from a Gaussian to the semicircle as m tends to infinity. We prove this convergence. We also prove that p_m is the product of a Gaussian and a certain even polynomial of degree 2m-2. The proof is by derivation of the moments from the eigenvalue trace formula. The key step is converting the central combinatorial problem in the calculation to an equivalent problem about Euler characteristic and algebraic topology. This is joint work with Gene Kopp.
• Andrew Ledoan (Boston College): Remark on a universality property of Gaussian analytic functions (with Marco Merkli, Shannon Starr): We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are independent and identically distributed, complex-valued random variables with mean zero and unit variance. Y. Peres and B. Virag have successfully shown that for the case of complex Gaussian coefficients, the zero set forms a determinantal point process with the Bergman kernel. Here we show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk.
• Lionel Levin (MIT): Logarithmic fluctuations from circularity: Start with n particles at the origin in the square grid Z², and let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a disk. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the disk, answering a question posed by Lawler in 1995. These logarithmic fluctuations were predicted numerically by chemical physicists in the 1980's. We also show that certain space-time averages of the fluctuations from circularity converge in law to a variant of the Gaussian free field.
• Steven J Miller (Williams): Painlev\'e VI and Tracy-Widom Distributions in Random Graphs, Random Matrix Theory and Number Theory: We report on two occurances of Painleve VI and Tracy Widom distributions. The first is in number theory and random matrix theory, where the observed repulsion near the central point of low-lying zeros of elliptic curve L-functions can be explained by a discretized random matrix ensemble, where the first eigenangle above 1 is given by a Painleve VI equation. The second involves the distribution of the second largest eigenvalue of d-regular graphs, which we show numerically is well-modeled by the beta = 1 Tracy-Widom distribution. If the observed growth rates of the mean and standard deviation as a function of the number of vertices holds in the limit, then in the limit approximately 52% (resp. 26%) of bipartite (resp. non-bipartite) d-regular graphs should be Ramanujan.
• Mihai Stoiciu (Williams): Random Matrices with Poisson Eigenvalue Statistics: We describe several classes of random matrices, both Hermitian and unitary, which exhibit local (microscopic) Poisson eigenvalue statistics. We describe the general strategy for proving these results and discuss connections between classes of random matrices with this property. We also present numerical evidence supporting the conjecture that the Poisson eigenvalue statistics holds for random non-Hermitian Anderson models (Hatano-Nelson matrices).

General: All the rooms have PCs or dual boot Macs connected to the internet and a projection system and black or white boards. Presenters could put their presentations on memory sticks or download them to the desktop from e-mail. Hotel accommodation information should be available around January 19th.

Schedule online at http://www.ams.org/meetings/sectional/2177_program_ss7.html#title

Questions? Email sjm1 AT williams.edu.