| Reconstruction
for Colorings on Trees. |
Nayantara Bhatnagar (UC Berkeley)
Consider a process where some information is propagated from the root
of a tree along the branches to all the leaves. During the
transmission, the information is modified by a random process at each
vertex. The question of reconstruction is whether the information
available at the leaves give any significant information about the
information at the root. The question arises naturally in a number of
areas including transmission of genetic information in a tree of
ancestors, noisy communication networks on trees, information theory
and statistical physics.
We will survey what is known about the problem, questions that remain
unresolved, and present some recent work (joint with Juan Vera and Eric
Vigoda) on the reconstruction problem for proper colorings of
trees.
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| Stationary Measures for Ballistic Annihilation with Branching. |
Maxim Krikun
(Institut Elie Cartan, Universite Henri Poincare)
A basic model of ballistic annihilation consists of a system of
particles in one-dimensional space, moving in either direction with
constant speed and annihilating upon collision, so that eventually most
of the particles disappear.
When branching is introduced into the picture, the system becomes
stable and admits a non-trivial stationary measure. Also, the union of
trajectories of all particles, seen as a subset of the plane, has some
interesting properties.
This is a joint work with Seruei Popov (USP)
and Philippe Chassaing and Lucas Gerin (IECN).
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| Dynamical Relaxation of a 1D Pinning Model. |
Fabio Martinelli
(University of Rome III)
We consider paths of a one–dimensional simple random walk conditioned
to come back to the origin after L
steps, In the pinning model each path
has a weight exp(c N(η)), where
N is the number of zeros in the path and c, the
pinning strength can be positive or
negative. When
the paths are constrained to be
non–negative, the polymer is said to satisfy a hard–
wall constraint. Such models are well known
to undergo a localization/delocalization
transition as the pinning strength is
varied. In this paper we study a natural "spin
flip" dynamics for these models, derive
several estimates on its spectral gap and
mixing time and discuss a possible signature
of the phase transition on the dynamics.
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| From
Markov Chains to Gaussian Priors and Back. |
Kshitij Khare
(Stanford)
The Dynkin isomorphism associates a Gaussian field to a Markov chain.
These Gaussian fields can be used as priors for prediction and time
series analysis. The older construction gives fields with all positive
covariances. A parallel construction of Diaconis-Evans gives fields
with all covariances negative. This work is extended to allow general
covariance sign patterns.
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| Normal Approximation with Continuous Symmetries. |
Elizabeth Meckes (Case Western Reserve University)
I
will describe a modification of Stein's method of exchangeable pairs
for normal approximation, for situations in which a random variable is
invariant under the action of a continuous group of symmetries. I
will discuss two applications of the method: firstly, to the
distributions of eigenfunctions of the Laplacian on compact Riemannian
manifolds, and secondly, to approximation of random orthogonal and
unitary matrices by Gaussian matrices.
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| Stochastic Differential Delay Equations. |
Matina Rassias (UC Berkeley)
Khasminskii (1969), introduced a powerful test for stochastic
differential equations (SDEs) to have non-exploding solutions without
satisfying a linear growth condition. Mao (2002), extended the above
idea for stochastic differential delay equations (SDDEs). We (with
our PhD dissertation, 2008) investigated an even more general
Khasminskii-type test for SDDEs which covers a wide class of
highly non-linear SDDEs and presented some moment estimates and
almost sure asymptotic estimates in order to study the long-term
behavior of the solution. In our talk we discuss some of the
above mentioned results.
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| Elliott-Kalton Stochastic Differential Games Associated with the Infinity Laplacian. |
Amarjit Budhiraja (UNC Chapel Hill)
Let G be a bounded domain in Rm and let
,
be given continuous functions. In a recent work,
Peres, Schramm, Sheffield, Wilson [PSSW] have considered a two player, zero sum, discrete time stochastic game, called
Tug of War. In this game, at each time instant
, one of the two players is selected by toss of a fair coin
who is then allowed to move the state
Xk
by an amount that is bounded by ε
.
The game ends at the first time instant
when Xk
with a payoff of
g(Xk)-(ε2/2)Σj=0k-1h(Xj).
Player 1 seeks to maximize this payoff while Player 2 aims to
minimize it. It is shown in [PSSW] that if inf h > 0 then the game has a value uε and as
ε,
uε converges uniformly to the "continuum value" u
which is the unique viscosity solution of the equation:
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Here
In this work we consider a continuous time two player zero sum stochastic differential game that is motivated
by the Tug of War game of [PSSW]. The state dynamics are driven by a one dimensional Brownian motion and
each player can control both drift and diffusion coefficients. In particular, diffusion control term can be unbounded and
possibly lead to degenerate dynamics. We show that the game has a value in the usual Elliott-Kalton sense which
is characterized as the unique viscosity solution of (1). Thus the result provides a game theoretic interpretation
of the "continuum value"
in the [PSSW] analysis.
This is a joint work with Rami Atar.
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| On the Expected Value of an L_2-bounded Martingale |
Isaac Meilijson (Tel Aviv University)
It
is shown that the ratio between the expected diameter of an L_2-bounded
martingale and the standard deviation of its last term cannot exceed
sqrt{3}. A quantity related to diameter, maximal drawdown (or rise), is
introduced and its expectation is shown to be bounded by sqrt{2} times
the standard deviation of the last term of the martingale. These
results complement the Dubins & Schwarz respective bounds 1 and
sqrt{2} for the ratios between the expected maximum and maximal
absolute value of the martingale and the standard deviation of its last
term. All four bounds are sharp.
Authors: Lester E. Dubins, David Gilat, Isaac Meilijson
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| The Alexander-Orbach Conjecture Holds in High Dimensions
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Asaf Nachmias (UC Berkeley)
It is known that the simple random walk on the unique infinite cluster
of supercritical percolation on Z^d diffuses in the same way it does on
the original lattice. In critical percolation, however, the behavior of
the random walk changes drastically.
The infinite incipient cluster (IIC) of percolation on Z^d can be
thought of as the critical percolation cluster conditioned on being
infinite. Alexander and Orbach (1982) conjectured that the spectral
dimension of the IIC is 4/3. This means that the probability of an
n-step random walk to return to its starting point scales like n^{-2/3}
(in particular, the walk is recurrent). In this work we prove this
conjecture when d>18; that is, where the lace-expansion estimates
hold.
Joint work with Gady Kozma.
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| Mixing on Random Graphs
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Allan Sly (UC Berkeley)
Understanding
Gibbs measures on trees and their spatial mixing thresholds
(uniqueness, strong spatial mixing, reconstruction) can give insight
into the behavior of Gibbs measures on general graphs, in particular
random graphs which are locally tree-like. We discuss a number of
cases where tree thresholds determine / does not determine the spatial
and temporal mixing properties of Gibbs measures on general and random
graphs. Partially based on joint work with E. Mossel.
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| Spatial Random Networks |
David Aldous (UC Berkeley)
I
will give an overview of ongoing research concerning networks linking
random vertices in two-dimensional space, emphasising two aspects.
(1) If we get to choose edges, subject to a given total length, then
how efficient can we make the network in the sense of providing routes
between vertices whose route-length is not much larger than
straight-line distance, and what precise statistic is most useful for
measuring this notion of ``efficient"?
(2) There is a general class of proximity graphs,
defined for arbitrary vertices, which are always connected.
Applying to random points gives a class of random networks which
are connected and have bounded mean degree. This class has
scarcely been studied, but seems an appealing modeling alternative to
the classical geometric random graphs for which one cannot have both connectivity and bounded mean degree.
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| Commutation Relations and Markov Chains
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Jason Fulman (USC)
It is shown that the combinatorics of commutation relations can be used
to give sharp convergence rate results for certain Markov chains. The
main example described in this talk will be a random walk on partitions
whose stationary distribution is the Ewens distribution from population
genetics.
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| The Lambda-coalescent Speed of Coming Down from Infinity
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Vlada Limic
(CNRS)
Consider a Lambda-coalescent that comes down from infinity, or
equivalently, that starts from a configuration containing
infinitely many blocks at time 0 and attains a configuration containing
a finite number N_t of blocks at any time t>0, almost surely. We
exhibit a deterministic function v : (0,∞) → (0,∞), such that N_t/v(t)
→ 1, almost surely and in L^p for any p>= 1, as t → 0. Our approach
relies on martingale methods. Based on a joint work with Julien and
Nathanael Berestycki.
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