Physically inspired stochastic partial differential equations
Anna Amirdjanova (U. Michigan at Ann Arbor)
The talk is devoted to the study of three fundamental equations of
mathematical physics (heat, Burgers' and Navier-Stokes) when the
classical models are perturbed by random forces. Questions of existence,
uniqueness and regularity of solutions will be discussed and some open
problems presented. In the second half of the talk we will focus on the
stochastic vorticity model, arising in hydrodynamics via stochastic point
vortex methods, and discuss its properties.
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Beta Tracy-Widom Laws, Random Schroedinger, and Diffusion
Brian Rider (U. Colorado at Boulder)
The beta ensembles of Random Matrix Theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (as well as their Wishart counterparts), these classical cases corresponding to beta = 1, 2, or 4. We prove that the largest eigenvalues of these general ensembles have limit laws described by the low lying spectrum of certain random Schroedinger operators. This provides a new characterization of the celebrated Tracy-Widom laws (now for all beta). As a corollary, there is a second, and also new, characterization in terms of the explosion probability of an associated one-dimensional diffusion. These descriptions even have applications, allowing for information on the shape of the "beta Tracy-Widom" distributions.
(Based on work with J. Ramirez and B. Virag.)
Extreme Values of Stable Random Fields and Long Range Dependence
Parthanil Roy (Cornell)
This talk will focus on the extreme values of stationary
symmetric \alpha-stable random fields over hypercubes of increasing size.
We will discuss how the asymptotic behavior of these extremes can be
connected to certain ergodic theoretical and group theoretical properties
of the integral representation of the random field. This connection helps
us to understand the long range dependence of such processes, which is
otherwise difficult to analyze due to absence of correlations. (This is a
joint work with Gennady Samorodnitsky.)
Diffusions in random environment and ballistic behavior
Tom Schmitz (UCLA)
We study continuous diffusions in random environment in d-dimensional space.
Once the environment is chosen, it remains fixed in time.
To restore some stationarity, it is common to average both with
respect to the path and environment mesure. One then obtains the so-called
annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random
environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting
In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law
of large numbers and a central limit theorem with non-degenerate covariance
As an application of our results, we highlight condition (T) as a source of
new examples of ballistic diffusions in random environment.
The diameter and mixing time of critical random graphs
Asaf Nachmias (Berkeley)
Let C1 denote the largest connected component of the critical
Erdos-Renyi random graph G(n,1/n). We show that, typically,
the diameter of C1 is of order n1/3 and the mixing time of the lazy
simple random walk on C1 is of order n. The latter answers a question of
Benjamini, Kozma and Wormald.
These results extend to clusters of size n2/3 of p-bond percolation
on any d-regular n-vertex graph where such clusters exist, provided
that p(d - 1) < 1 + O(n-1/3).
Joint work with Yuval Peres.
Spectrum of Levy matrices
Alice Guionnet (ENS de Lyon & Berkeley)
Take a square N × N symmetric
matrix XN with real i.i.d random entries
above the diagonal with law &mu such that &mu(x2) = 1. Then, if
&lambda1,&hellip,&lambdaN denote the eigenvalues of XN,
Wigner's theorem asserts that the spectral
measure N-1&sum &deltaN-1/2&lambdai
converges weakly in expectation and almost surely
towards the semi-circle law.
We study what happens when &mu has no finite second moment,
but belong to the domain of attraction
of an &alpha-stable law. In particular,
we show the weak convergence of E[N-1&sum &deltaN-1/&alpha&lambdai] towards a symmetric law with heavy
Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models
Charles Bordenave (Berkeley)
We will study the relation between the minimal spanning tree (MST) on
many random points and the "near-minimal" tree which is optimal
subject to the constraint that a proportion \delta of its edges must
be different from those of the MST. Heuristics suggest that,
regardless of details of the probability model, the difference of the
lengths should scale as the number pf points times \delta squared.
We will see why this scaling result is true in the model of the
lattice with random edge-lengths. In the 2-dimensional Euclidean
model, by exploiting the well-known connection between MSTs and
continuum percolation of the Gilbert disc model we will reduce the
scaling result to an Ansatz that a known technical result for lattice
percolation extends to continuum percolation.
This Ansatz will be related to Kesten's work on the critical exponents
in two dimensional percolation.
This is a joint work with David Aldous (UC Berkeley) and Marc Lelarge
(Ecole Normale Superiere and INRIA)
Measure Concentration of Strongly Mixing Processes with Applications
Leonid Kontorovich (CMU)
The concentration of measure phenomenon was first discovered in 1930's
by Paul Levy and has been investigated since then, with increasing
intensity in recent decades. The probability-theoretic results have
been gradually percolating throughout the mathematical community,
finding applications in Banach space geometry, analysis of algorithms,
statistics and machine learning.
There are several approaches to proving concentration of measure
results; we shall offer a brief survey of these. The principal
contribution of this work is a new concentration inequality for
nonproduct measures. The inequality is proved by elementary means,
yet enables one, with minimal effort, to recover and generalize the
best current results for Markov chains, as well as to obtain
(apparently) new results for hidden Markov chains and Markov trees.
Applications of these inequalities include a strong law of large
numbers for a broad class of non-independent processes. In particular,
this allows one to analyze the convergence of inhomogeneous Markov
Chain Monte Carlo algorithms. We have also obtained some partial
results on extending the Rademacher-type generalization bounds to
processes with arbitrary dependence.
Ergodicity in infinite dimensions: SPDEs and the Wasserstein metric
Jonathan Mattingly (Duke)
I will discuss a particular perspective on proving ergodic theorems for
infinite dimensional Markov processes. I will begin with finite
dimensional systems and work my way up to infinite dimensional systems.
Specifically, I will give a version of Harris's
ergodic result in the Wasserstein metric (*not* total variation). I
will talk about some recent results in the ergodic theory of
stochastic partial differential equations. I will
then discuss some tools from Malliavin calculus used in proving the
Spectral Norm of Random Wigner Matrices with Non-Symmetrically Distributed Entries
Alexander Soshnikov (UC Davis)
In the first part of the talk, I will show that the spectral radius of an N × N random symmetric matrix with i.i.d. bounded centered but non-symmetrically
distributed entries is bounded from above by 2&sigma + o(N-6/11+&epsilon),
where &sigma2 is the variance of the matrix
entries and &epsilon is an arbitrary small positive number. Our bound improves the earlier results by Z. Füredi and
J. Komlós (1981), and Van Vu (2005). Our approach heavily relies on combinatorial considerations. This is a joint paper with Sandrine Peche.
In the second part of the talk, I will discuss a resolvent approach to study the universality at the edge of the spectrum.
Probability and Spatial Networks
David Aldous (Berkeley)
Network design and analysis have been studied in many different applied
contexts, yet many simple-to-state abstracted mathematical problems
have not been studied very systematically. For a road network on n
cities, what is the trade-off between total network length and the
efficiency of the network in providing short routes? For an airline
network on n cities, requiring routes to have an average of no more
that 3 hops, how short can network length be? Such questions can
involve probability in several ways. First, the ``average case"
model of randomly-distributed cities is a natural counterpart to
worst-case analysis. Second, while upper bounds on performance are
obtained by explicit construction, lower bounds need more
mathematical arguments provided by classical integral geometry.
Third, the Poisson line process turns out to be very useful!
[Joint work with Wilf Kendall. The two papers discussed are available at http://arxiv.org/abs/cond-mat/0702502
and http://front.math.ucdavis.edu/math.PR/0701140 ]
A Multivariate Statistical Approach to Performance Analysis of Wireless Communication Systems
Siamak Sorooshyari (Lucent Technologies - Bell Labs)
The explosive growth of wireless communication technologies has placed paramount importance on accurate
performance analysis of the fidelity of a service offered by a system to a user. Unlike the channels of
wireline systems, a wireless medium subjects a user to time-varying detriments such as multipath fading,
cochannel interference, and thermal receiver noise. As a countermeasure, structured redundancy in the form
of diversity has been instrumental in ensuring reliable wireless communication characterized by a low bit
error probability (BEP). In the performance analysis of diversity systems the common assumption of
uncorrelated fading among distinct branches of system diversity tends to exaggerate diversity gain resulting
in an overly optimistic view of performance. A limited number of works take into account the problem of
statistical dependence. This is primarily due to the mathematical complication brought on by relaxing the
unrealistic assumption of independent fading among degrees of system diversity.
We present a multivariate statistical approach to the performance analysis of wireless communication systems
employing diversity. We show how such a framework allows for the statistical modeling of the correlated
fading among the diversity branches of the system users. Analytical results are derived for the performance
of maximal-ratio combining (MRC) over correlated Gaussian vector channels. Generality is maintained by
assuming arbitrary power users and no specific form for the covariance matrices of the received faded
signals. The analysis and results are applicable to binary signaling over a multiuser single-input
multiple-output (SIMO) channel. In the second half of the presentation, attention is given to the
performance analysis of a frequency diversity system known as multicarrier code-division multiple-access
(MC-CDMA). With the promising prospects of MC-CDMA as a predominant wireless technology, analytical results
are presented for the performance of MC-CDMA in the presence of correlated Rayleigh fading. In general, the
empirical results presented in our work show the effects of correlated fading to be non-negligible, and most
pronounced for lightly-loaded communication systems.
Hierarchical beta processes and the Indian buffet process
Romain Thibaux (Berkeley)
The Indian buffet process is an exchangeable process that generates
binary matrices. It has been used in machine learning applications as
a prior over models where objects can belong to overlapping categories
(factorial models). I will show its close relationship with the beta
process, a nonparametric prior originally developed for survival
analysis. For people familiar with the Dirichlet process, the beta
process and the Indian buffet process are close analogs of the
Dirichlet process and the Chinese restaurant process. I will explain
and show examples of all these objects. I will then use this link to
derive algorithms and define hierarchies of beta processes.
Spin glasses and Stein's method
Sourav Chatterjee (Berkeley)
The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.
On the localization transition for pinning models
Giambattista Giacomin (Université Paris 7)
Pinning models and are Gibbs measures built on
renewal processes. Alternatively they can be seen
as one dimensional statistical mechanics systems
with long tail two-body interactions. This class of models
includes and gives a unitary vision of a large number of systems
modeling phenomena like polymer pinning on a defect line,
interface wetting, DNA denaturation, and more.
We will review basic features of pinning models
and we will present some recent progress, notably in
the analysis of the behavior of the correlation length
of the system near criticality.