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Session: Probability, Combinatorics, and Statistical Mechanics
Organizer: Russell Lyons, Indiana University
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| Speaker: |
Scott Sheffield, Courant Institute and IAS |
| Title: |
Tug of war and the infinity Laplacian |
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Abstract:
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The infinity Laplacian (informally, the "second derivative in
the gradient direction") is a simple yet mysterious operator with many
applications.
"Tug of war" is a two player random turn game played as follows:
SETUP: Assign each player one of two disjoint target sets T_1 and T_2 in
the plane, and fix a starting position x and a constant epsilon. Place
the game token at x.
GAME PLAY: Toss a fair coin and allow the player who wins the coin toss to
move the game token up to epsilon units in the direction of his or her
choice. Repeat the above until the token reaches a target set T_i. The
ith player is then declared the winner.
Given parameters epsilon and x, write u_epsilon(x) for the probability
that player one wins when both players play optimally. We show that as
epsilon tends to zero, the functions u_epsilon(x) converge to the infinity
harmonic function with boundary conditions 1 on T_1 and 0 on T_2.
Our strategic analysis of tug of war leads to new formulations and
significant generalizations of several classical results about infinity
laplacians. The game theoretic arguments are simpler and more elementary
than the original proofs.
This talk is based on joint work with Yuval Peres, Oded Schramm, and David
Wilson.
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| Speaker: |
Antal Jarai, Carleton University |
| Title: |
Infinite volume limit of the Abelian sandpile model on Zd |
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Abstract:
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The Abelian 'sandpile' model was introduced by physicists
as a basic example of self-organized criticality (SOC). Roughly
speaking, SOC arises when a stochastic dynamics drives a system
towards a stationary state characterized by power laws. We study
existence of the infinite volume limit for the model, and properties
of this limit, giving insight into the asymptotic behaviour of
large 'sandpiles'. Most progress can be made above the upper critical
dimension d > 4. We discuss some open problems related to extending
these results to lower dimensions.
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| Speaker: |
Richard Kenyon, University of British Columbia |
| Title: |
Simple random surfaces |
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Abstract:
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This is joint work with David Brydges and Jessica Young. We
study a model of random surfaces coming with an immersion into an
arbitrary two-complex. Certain probabilistic quantities can be computed
using the Green's function for the Laplacian on 1-forms. |
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Session: Flows and Random Media
Organizer: Mike Cranston, University of California, Irvine
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| Speaker: |
Timo Seppalainen, University of Wisconsin |
| Title: |
Spatial inhomogeneities and large scale behavior of the asymmetric exclusion process |
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Abstract:
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This talk describes some results and open problems
for the one-dimensional asymmetric exclusion process in
situations where spatial inhomogeneities, either deterministic
or random, are added to the model. |
| Speaker: |
Peter Mueller, University of Goettingen |
| Title: |
Spectral asymptotics of Laplacians on bond-percolation graphs |
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Abstract:
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Bond-percolation graphs are random subgraphs of the
d-dimensional integer lattice generated by a standard Bernoulli
bond-percolation process. The associated graph Laplacians, subject to
Dirichlet or Neumann conditions at cluster boundaries, represent bounded,
self-adjoint, ergodic random operators. They possess almost surely the
non-random spectrum [0,4d] and a self-averaging integrated density of
states. This integrated density of states is shown to exhibit Lifshits
tails at both spectral edges in the non-percolating phase. Depending on
the boundary condition and on the spectral edge, the Lifshits tail
discriminates between different cluster geometries (linear clusters versus
cube-like clusters) which contribute the dominating eigenvalues. Lifshits
tails arising from cube-like clusters continue to show up above the
percolation threshold. In contrast, the other type of Lifshits tails
cannot be observed in the percolating phase any more because they are
hidden by van Hove singularities from the percolating cluster. |
| Speaker: |
Ken Alexander, University of Southern California |
| Title: |
The pinning transition for a polymer in the presence of a random
potential |
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Abstract:
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We consider a polymer, with monomer locations modeled by the
trajectory of a Markov chain, in the presence of a potential that
interacts with the polymer when it visits a particular site 0. We assume
the probability of an excursion of length $n$ from 0, in the absence of
the potential, decays like $n^{-c}$ for some $c>1$. Disorder is introduced
by, having the interaction vary from one monomer to another, as a constant
$u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above
which the polymer is pinned, placing a positive fraction, called the
contact fraction, of its monomers at 0 with high probability. We obtain
bounds for the contact fraction near the critical point and examine the
effect of the disorder on the specific heat exponent, which describes the
approach to 0 of the contact fraction at the critical point. Our results
are consistent with predictions in the physics literature that the effect
of disorder is quite different in the cases $c<3/2$ and $c>3/2$. |
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Session: Stochastic Integration
Organizer: Terry Lyons, Oxford University
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| Speaker: |
Peter Friz, Cambridge University |
| Title: |
Anticipating stochastic calculus via good rough path sequences |
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Abstract:
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We consider anticipative Stratonovich stochastic differential
equations driven by some stochastic process (not necessarily a
semi-martingale). No adaptedness of initial point or vector fields is
assumed. Under a simple condition on the stochastic process, the unique
solution of the above SDE understoof in the rough path sense is
actually a Stratonovich solution. This condition is satisfied by Brownian
motion and fractional Brownian motion with Hurst parameter greater than
1/4. |
| Speaker: |
Anastasia Papavasiliou, Princeton University |
| Title: |
Applications of rough paths to speech recognition |
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Abstract:
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It is a well known fact that when someone speaks, the signal
reaching the human ear is the original signal produced by the speaker and
several delays. We think of this as a multidimensional rough path. Using
tools coming from the theory of rough paths, I will construct much simpler
rough paths (piecewise linear) approximating the one containing the speech
signal and its delays, which still cause a similar response. Thus, the
constructed rough paths contain all the information relevant to the
response. If we think of the "meaning" of the speech signal as the
response, the constructed rough paths will contain the "meaning" in a much
more robust way than the signal itself, and thus can be used to construct
a likelihood function for each word. |
| Speaker: |
Zhongmin Qian, Oxford University |
| Title: |
Stochastic integrals for processes with long-time memory |
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Abstract:
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Stochastic processes with long-time memory have been used
extensively in modelling random phenomena. In this talk I will discuss the
theory of rough paths and its application to a class of stochastic
dynamical systems driven by such long-time memory. |
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Session: Random Walk in Random Environment
Organizer: Ofer Zeitouni, University of Minnesota
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| Speaker: |
Nina Gantert, University of Muenster |
| Title: |
Random walk in random scenery |
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Abstract:
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Let $(Z_n)_{n\in N_0}$ be a $d$-dimensional random walk in
random scenery, i.e., $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in N_0}$
a random walk in $Z^d$ and $(Y(z))_{z\in Z^d}$ an i.i.d. scenery,
independent of the walk. The walker's steps have mean zero and finite
variance.
We identify the speed and the rate of the logarithmic decay of $P(\frac 1n
Z_n>b_n)$ for various choices of sequences $(b_n)_n$ in $[1,\infty)$.
Depending on $(b_n)_n$ and the upper tails of the scenery, we identify
different regimes for the speed of decay and different variational
formulas for the rate functions.
In contrast to recent work by A.~Asselah and F.~Castell, we consider
sceneries unbounded to infinity. It turns out that there are interesting
connections to large deviation properties of self-intersections of the
walk, which have been studied recently by X. Chen. The talk is based on
joint work with Wolfgang Koenig, Remco van der Hofstad and Zhan Shi.
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| Speaker: |
Vladas Sidoravicius, IMPA |
| Title: |
Growth in dynamic random environment |
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Abstract:
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We consider the following model for the spread of an infection:
There is a ``gas'' of so-called A-particles, each of which performs a
continuous time simple random walk on Z^d, with jumprate D_A. We assume
that initially the number of A-particles at x is independent, mean mu_A
Poisson distribution. In addition, there are B-particles which perform
continuous time simple random walks with jumprate D_B. Initially we start
with only one B-particle in the system, located at the origin. The
B-particles move independently of each other, and the only interaction is
that when a B-particle and an A-particle coincide, the latter
instantaneously turns into a B-particle. For different values of the
parameters D_A and D_B one obtains several interesting evolutions, raging
from stochastic sandpile type dynamics (activated random walkers) to
contact process like evolution and DLA-type growth. The difficult aspect
of all these models is the absence of useful subadditive quantities. I
briefly discuss these models before going to our main results: Consider
the set C(t):= {x in Z^d: a B-particle visits x during [0,t]}. If D_A=D_B,
then B(t) := C(t) + [- 1/2, 1/2]^d grows linearly in time with an
asymptotic shape, i.e., there exists a non-random set B_0 such that
(1/t)B(t) \to B_0, in a sense which will be made precise. Moreover, if the
"recuperation'' transition form A to B occurs at the rate lambda>0, for
each particle independently, then we show that there is a phase transition
between survival and extiction of B particles. I also briefly present the
key ideas of the proof.
The talk is based on joint works with H. Kesten.
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| Speaker: |
Martin Zerner, University of Tuebingen |
| Title: |
On some self-interacting random walks in random environment |
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Abstract:
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TBA |
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Session: Stochastic Partial Differential Equations
Organizer: Jonathan Mattingly, Duke University
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| Speaker: |
Martin Hairer, University of Warwick |
| Title: |
Stochastic modulation equations |
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Abstract:
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We consider the stochastic Swift-Hohenberg equation on a large
domain near its change of stability. We show that, under the appropriate
scaling, its solutions can be approximated by a periodic wave, which is
modulated by the solutions to a stochastic Ginzburg-Landau equation.
Unlike in the deterministic case, this approximation holds for all times
and extends to the respective invariant measures.
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| Speaker: |
Jonathan Mattingly, Duke University |
| Title: |
Ergodicity of the degenerately forced stochastic fluid equations |
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Abstract:
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I will present a theory which allows one to prove the uniqueness
of the stationary measure for a large class of SPDEs with additive noise.
To do so, I will discuss Malliavin Calculus in the SPDE setting and a
generalization of Hormander's hypo-elliptic theory to the SPDEs. I will
also discuss a new generalization of the strong Feller property which
seems to be use full in infinite dimensions. |
| Speaker: |
Nicolai Krylov, University of Minnesota |
| Title: |
On the foundation of the Lp-theory of SPDEs |
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Abstract:
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We discuss a detailed proof of a generalization of the
Littlewood-Paley inequality upon which the Lp-theory of SPDEs is
based. |
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Invited Lectures
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| Speaker: |
Terry Lyons, Oxford University |
| Title: |
Rough paths: a top down description of controls |
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Abstract:
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The theory of rough paths as developed by the Author (and several others including those speaking at this workshop as well as others such as Hambly, Ledoux, Coutin, ..) aims to study the differential equations used to model the situation where a system responds to external control or forcing. The theory describes a robust approach to these equations that allows the forcing to be far from differentiable. The methodology permits the main probabilistic classes as well as many new types of stochastic forcing that do not fit into the classical semi-martingale setting directly.
The key to this theory is to answer the question - when do two controls
produce similar responses. This is also a core question for the problem
for multi-scale analysis where one needs to summarise small scale
behaviour in a way that large scale responses can be predicted from the
summarised information. The question can be translated into one asking
that one characterises the continuity properties of the It? map. This is
indeed possible and the Universal Limit theorem proves the (uniform)
continuity of the map taking the forcing control to response for a wide
class of metrics on smooth paths - and the completions of the space under
these metrics give the so called rough paths - giving insight into the
control problem.
The approach is quite structured, and allows one to give a top down
analysis of a control in terms of a sequence of algebraic coefficients we
call the signature of the control (which have similarity to a child's
pr?cis of a complicated text by a simpler one and are a non-commutative
analogue of Fourier coefficients) with refinements giving more accurate
information about the control. Hambly and Lyons recently proved that this"
signature" of a control completely characterises the control up to the
appropriate null sets.
The talk will summarise some of this work.
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| Speaker: |
Russell Lyons, Indiana University |
| Title: |
Unimodularity and stochastic processes |
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Abstract:
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Stochastic processes on vertex-transitive graphs, especially Cayley graphs of groups, have been studied for 50 years (not counting the special case of integer lattices, which goes back hundreds of years). The assumption of invariance under graph automorphisms plays a key role, but investigations of the last 15 years have shown that an additional assumption is also extremely useful. This newer assumption is the property of unimodularity, which is equivalent to the Mass-Transport Principle. We shall review some well-known applications and also discuss recent work with David Aldous. |
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Medallion Lectures
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| Speaker: |
Ofer Zeitouni, University of Minnesota |
| Title: |
Recent results and open problems concerning motion in random media |
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Abstract:
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The talk will describe the current knowledge and some of the challenges in the study of motion in random environment, focusing on the model of random walk in random environment and the related model of diffusions in random environments. For the latter, I will describe a CLT in the case where the environment is a small perturbation of the constant environment. |
| Speaker: |
Amir Dembo, Stanford University |
| Title: |
The disconnection time of the random walk on a discrete cylinder |
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Abstract:
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Consider the simple random walk on the discrete cylinder whose
base is the d-dimnesional torus with side-length N, and whose height is
the set of all integers. When d>1, the time the walk needs to disconnect
the discrete cylinder is very roughly of order N to the power 2d, and
comparable to the cover time of the slice of height 0. Further, by the
time disconnection occurs, a massive "clogging'' takes place in the
truncated cylinders of height N to power d' for any d'<d.
I shall also describe what we know about the disconnection
time for base graphs other than the d-dimensional torus.
This talk is based on a joint work with Alain-Sol Sznitman.
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