Complexity of random Morse functions of many variables.
I. Link with Random matrices |
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Gerard Ben Arous (Courant Institute/NYU)
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| II. Low energy states of spin glasses and Parisi's replica symmetry breaking
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Gerard Ben Arous (Courant Institute/NYU)
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| Fluctuations and Contours of the Ginzburg-Landau Model on a Bounded Domain
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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).
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| Universality of Random Matrices and Dyson Brownian Motion
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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
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| On competing particle systems
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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.
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| A Coupling Argument for the Random Tranposition Walk on S_n
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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.
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| Markov chain approximations to non-symmetric diffusions in divergence form with bounded coefficients
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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.
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| Counting the Faces of Randomly Projected Hypercubes:
Phase transitions and applications
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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).
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| Large deviations for random walk in a random environment
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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.
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| On the contraction method for convolution equations
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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.
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| Dimers and analytic torsion
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Julien Dubedat (Columbia)
We discuss Gaussian invariance principles for dimer models in relation
with variational formulae for zeta-determinants of Cauchy-Riemann
operators.
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