Stanford Probability Seminar Mondays, 4:15 - 5:15pm (Refreshments at 4pm in the 1st floor lounge) Sequoia Hall, Room 200 To edit your subscription to the mailing list, click here and scroll down Contact Slava Kargin (kargin@math.stanford.edu) for organizational matters.

# Winter Quarter 2010

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Date Speaker Title
(click for abstract)
7 Jan Gerard Ben Arous (Courant Institute/NYU) Complexity of random Morse functions of many variables.
Bergman Lecture. Building 420. Room 41. Reception at 3pm at Sloan Hall lounge.
8 Jan Gerard Ben Arous (Courant Institute/ NYU) II. Low energy states of spin glasses and Parisi's replica symmetry breaking Bergman Lecture. Sloan Hall, Room 380-C. Dinner
11 Jan Jason Miller (Stanford) Fluctuations and Contours of the Ginzburg-Landau Model on a Bounded Domain
18 Jan No Seminar Holiday
19 Jan Horng-Tzer Yau (Harvard) Universality of Random Matrices and Dyson Brownian Motion joint with Applied Mathematics seminar, Tuesday, 4:15 in 380-X
25 Jan Mykhaylo Shkolnikov (Stanford) On competing particle systems
1 Feb Olena Bormashenko (Stanford) A Coupling Argument for the Random Tranposition Walk on S_n
8 Feb Jean-Dominique Deuschel (Technische Universität Berlin) Markov chain approximations to non-symmetric diffusions in divergence form with bounded coefficients Dinner
15 Feb No Seminar Holiday
22 Feb David Donoho (Stanford) Counting the Faces of Randomly Projected Hypercubes: Phase transitions and applications
1 Mar Atilla Yilmaz (Berkeley) Large deviations for random walk in a random environment Dinner
8 Mar Erwin Bolthausen (Institut für Mathematik Universität Zürich) On the contraction method for convolution equations Dinner
15 Mar Julien Dubédat (Columbia) Dimers and analytic torsion Unusual place: Math building, room 383N; Dinner

# Abstracts

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 Complexity of random Morse functions of many variables. I. Link with Random matrices Gerard Ben Arous (Courant Institute/NYU) top of page II. Low energy states of spin glasses and Parisi's replica symmetry breaking Gerard Ben Arous (Courant Institute/NYU) top of page Fluctuations and Contours of the Ginzburg-Landau Model on a Bounded Domain Jason Miller (Stanford) The object of this study is the massless field on D_n, the intersection of a two-dimensional bounded domain D and the integer lattice scaled by 1/n. The Hamiltonian is $$H(h) = \sum_{x \sim y} V(h(x) - h(y))$$ and the boundary condition h(x) = f(x) for a given continuous function from Z^2 to Z. The interaction V is assumed to be symmetric, strictly convex, and have bounded second derivatives. This is a general model for a $(2+1)$-dimensional effective interface. In the talk we will discuss two new limit theorems for this model. Firstly, linear functionals of h converge in the limit to a Gaussian free field on D. Secondly, the level curves are asymptotically described by SLE(4). top of page Universality of Random Matrices and Dyson Brownian Motion Horng-Tzer Yau (Microsoft) The universality for eigenvalue spacing distributions is a central question in the random matrix theory. In this talk, we introduce a new general approach based on comparing the Dyson Brownian motion with a new related dynamics, the local relaxation flow. This method can be applied to prove the universality for the eigenvalue spacing distributions for the symmetric, hermitian, self-dual quaternion matrices and the real and complex sample covariance matrices. A central tool in this approach is to estimate the entropy flow via the logarithmic Sobolev inequality top of page On competing particle systems Mykhaylo Shkolnikov (Stanford) Competing particle systems were introduced by Ruzmaikina and Aizenman (2005) in the context of the Sherrington-Kirkpatrick model of spin glasses. Their goal was to determine the stationary measures for the process of gaps in a certain discrete time evolution of points on the real line. I will give extensions of their result and talk about related continuous time evolutions recently introduced by Chatterjee and Pal (2007) and Pal and Pitman (2008). If time permits, I will discuss applications of these processes to financial mathematics and queueing theory. top of page A Coupling Argument for the Random Tranposition Walk on S_n Olena Bormashenko (Stanford) I will present a Markovian coupling argument that gives the correct order mixing time for the random transposition walk on S_n. This walk is defined as follows: at each step, we choose i and j independently from {1,2,...,n} and transpose them (possibly, i = j, in which case the walk stays in place.) This walk is known to mix in O(n log n) time: however, in this form it does not admit a Markovian coupling that proves an O(n log n) mixing time for the walk. By lifting the random walk to conjugacy classes, and analyzing the resulting walk (called a split-merge random walk), I will be able to demonstrate a coupling that meets in O(n log n) time, and therefore provides the correct order bound on the mixing time of the random transposition walk on S_n. top of page Markov chain approximations to non-symmetric diffusions in divergence form with bounded coefficients Jean-Dominique Deuschel (Technische Universität Berlin) Consider a diffusion in divergence form with a uniformly elliptic and bounded non-symmetric diffusion matrix. We construct a sequence of random walks which converges to the diffusion process. Our method is based on heat kernel estimates for centered random walks. This is joint work with Takashi Kumagai. top of page Counting the Faces of Randomly Projected Hypercubes: Phase transitions and applications David Donoho (Stanford) Suppose we randomly-project an N-dimensional hypercube into n-dimensional space. The resulting zonotope in R^n has typically fewer k-faces than the original hypercube in R^N. How many fewer? We'll give the exact answer. It exhibits a phase transition: there is a function rho(delta) such that, for k < n rho(n/N)(1+o(1)) the two objects have about the same number of k-faces, but not otherwise. I'll try to make clear the connection with classical work of Tom Cover and Brad Efron. The applications are to Compressed Sensing. If we are trying to solve a system of linear equations with n equations and N unknowns, and the answer is known a priori in the hypercube, the phase transition curve explains precisely when there typically is a unique solution to the system within the hypercube. Roughly speaking we can typically recover a k-simple vector provided k < n rho(n/N), where k-simple means that at most k of the coordinates are not at the bounds {0,1}; we can do this by linear programming. This is joint work with Jared Tanner (Edinburgh). top of page Large deviations for random walk in a random environment Atilla Yilmaz (Berkeley) I will talk about large deviations for nearest-neighbor random walk in an i.i.d. environment on Z^d. There exist variational formulae for the quenched and the averaged rate functions I_q and I_a, obtained by Rosenbluth and Varadhan, respectively. I_q and I_a are not identically equal. However, when d>3 and the walk satisfies the so-called (T) condition of Sznitman, they are equal on an open set A_{eq}. Moreover, for every \xi in A_{eq}, there exists a positive solution to a Laplace-like equation involving \xi and the original transition kernel of the walk. This solution lets us define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at \xi. It also corresponds to the unique minimizer of Rosenbluth's variational formula provided that the latter is slightly modified. In other words, when the limiting average velocity of the walk is conditioned to be equal to \xi, the walk chooses to tilt its original transition kernel by an h-transform. On the contraction method for convolution equations Erwin Bolthausen (U. Zurich) We present a new version of the contraction method for the asymptotic behavior of solutions of convolution equations of the type appearing for instance in self-avoiding random walks. The method appeared first in a paper with Christine Ritzmann. The new version is simpler, and probably more flexible. This is work in progress with Christine Ritzmann and Felix Rubin. top of page Dimers and analytic torsion Julien Dubedat (Columbia) We discuss Gaussian invariance principles for dimer models in relation with variational formulae for zeta-determinants of Cauchy-Riemann operators. top of page
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