COLUMBIA-PRINCETON PROBABILITY DAY

                                                                                             MAY 1, 2009.

The conference will be held in Columbia University, Schermerhorn Hall room 501 (campus map).

Registration is free by RSVP to cpday@math.columbia.edu by April 22.

Poster


Speakers (preliminary):


Steve Evans (Berkeley)

Martin Hairer (Warwick/NYU)

Van Vu (Rutgers)

Francesco Cellarosi (Princeton)

Jason Miller (Stanford)

Cedric Villani (ENS Lyon)






Schedule


9-10 a.m. : registration/coffee

10-11a.m. : Martin Hairer "Ergodic theory of non-Markovian stochastic processes"

11-12 a.m. :  Steven Evans "Eigenvalues of large random trees"

12-2 pm.: lunch break

2-3 p.m. : Cedric Villani "Logarithmic Sobolev inequalities, concentration etc."

3-4 p.m. : Van Vu "Random matrices: The distribution of the smallest singular values (Universality at the Hard Edge)"

4-4.30 p.m. : Francesco Cellarosi "On the curlicue measure generated by quadratic trigonometric sums."

4-30-5 p.m. : Jason Miller "Thick Points of the Gaussian Free Field"

5-6:30 p.m. : wine & cheese reception


Abstracts



Martin Hairer.

Title: Ergodic theory of non-Markovian stochastic processes

Abstract. We consider evolution equations driven by a noise that is not white in time, so that the resulting process does not have the Markov property. We show that there is an analogue in this setting to the usual Doob-Khashminski ergodicity criterion, provided that the driving noise satisfies a certain "quasi-Markov" property. This can be verified in many cases, including SDEs driven by fractional Brownian motion and thus having long-range memory.



Steven N. Evans

Title. Eigenvalues of large random trees

Abstract. A common question in evolutionary biology is whether evolutionary processes leave some sort of signature in the shape of the phylogenetic tree of a collection of present day species. Similarly, computer scientists wonder if the current structure of a network that has grown over time reveals something about the dynamics of that growth. Motivated by such questions, it is natural to seek to construct ``statistics'' that somehow summarize the shape of trees and more general graphs, and to determine the behavior of these quantities when the graphs are generated by specific mechanisms. The eigenvalues of the adjacency and Laplacian matrices of a graph are obvious candidates for such descriptors. I will discuss how relatively simple techniques from linear algebra and probability may be used to understand the eigenvalues of a very broad class of large random trees. These methods differ from those that have been used thusfar to study other classes of large random matrices such as those appearing in compact Lie groups, operator algebras, physics, number theory, and communications engineering. This is joint work with Shankar Bhamidi (U. of British Columbia) and Arnab Sen (U.C. Berkeley).



Cedric Villani

Title. Logarithmic Sobolev inequalities, concentration etc.

Abstract. I will start by recalling the theorem which I obtained ten years ago with Felix Otto, relating logarithmic Sobolev inequalities with Talagrand's concentration inequalities, and describe a recent robust approach to this result by Nathael Gozlan, based on large deviations. Then I will describe recent work of Felix and I in collaboration with Maria Reznikoff-Westdickenberg and Natalie Grunewald, applying the principles of logarithmic Sobolev inequalities and concentration, to provide a transparent, quantitative, analytic proof of the Guo-Papanicolaou-Varadhan theorem on hydrodynamic limits of Ginzburg-Landau stochastic particle systems.


Van Vu

Title. Random matrices: the distribution of the smallest singular values (Universality at the Hard Edge)

Abstract. Let x be a real-valued random variable of mean zero and variance one. Let M(n) denote the random matrix of size n whose entries are iid copies of x and s(n) denote the least singular value of $M_n(x)$. (One can also view s(n)^2 as the least eigenvalue of the Wishart matrix M(n) M(n)*. ) The problem of understanding the distribution of the least singular value of a random matrix was first raised by von Neumann and Goldstine in the 1940s, in their studies on numerical linear algebra. Since then, it has become an important problem in the theory of random matrices, numerical analysis and smooth analysis of linear programming. Results for special case where x is gaussian or gaussian divisible were obtained by Edelman, Forrester, Ben Arous-Peche, Ramirez-Rider and others. We show that (under a finite moment assumption) the probability distribution of n^{1/2} s(n) is UNIVERSAL in the sense that it does NOT depend on the distribution of x. Our approach is guided by the general idea of ``property testing'' from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices. Joint work with T. Tao (UCLA).


Francesco Cellarosi

Title. On the curlicue measure generated by quadratic trigonometric sums.

Abstract. I shall discuss the ensemble of complex curves generated by quadratic trigonometric sums. The geometric multi-scale structure of such curves is naturally connected with infinite ergodic theory and is studied dynamically with the help of continued fractions with even partial quotients. A renewal-type limit theorem for these continued fractions, along with a renormalization scheme and some probabilistic tools, allows us to prove that these curves are distributed according to some limiting probability measure. This work generalizes some results by J. Marklof and Jurkat and van Horne.

Jason Miller.

Title: Thick Points of the Gaussian Free Field

Abstract: Let $U \subseteq \C$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U \nabla f(x) \cdot \nabla g(x) dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z \in U$ such that the average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi} \log \tfrac{1}{r}$ as $r \to 0$. We show that for each $0 \leq a \leq 2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$ and that with probability one $T(a;U)$ is empty when $a > 2$. Furthermore, we prove that $T(a;U)$ is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter $\gamma$ given formally by $\Gamma(dz) = e^{\sqrt{2\pi} \gamma F(z)} dz$ considered by Duplantier and Sheffield.