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Titles and Abstracts
- Peter Winkler
Permutation Limits and Pattern Densities
The space of permutations of 1,2,...,n for all n has a
natural completion to the space of probability
measures on the unit square with uniform marginals
(also known as doubly-stochastic measures, or
2-dimensional copulas). The "pattern density" of
a fixed permutation in a copula is the probability
that the relative orders of the x- and y-coordinates
of random points yield that permutation.
We want to know
not only which sets of pattern densities are
realizable, but what most permutations with given
densities look like. To do this we employ a
variational principle saying that the copula
representing most members of such a class of
permutations is the one maximizing a certain entropy
integral.
In work done last
spring at ICERM with Rick Kenyon (Brown), Dan Kral'
(Warwick) and Charles Radin (Texas), analysis and
heuristics have been used to make a dent in this large
set of problems.
- Rick Kenyon
Dimers and geometry
Given a 3-connected planar graph G, the space of
embeddings with convex faces and pinned boundary
is homeomorphic to a ball. We give global coordinates
and natural probability measures on this space.
We also discuss various subspaces of embeddings of
planar graphs, including those with fixed face shapes,
fixed edge directions, fixed face areas, and the
relation of all of these concepts with the Kasteleyn
matrix.
- Allan Sly
Counting Solutions to Random Constraint
Satisfaction Problems
Satisfaction problems subject to random constraints
are a well-studied area in combinatorics and the
theory of computation, for example random colourings
of random graphs and random k-sat. Ideas from
statistical physics provide a detailed description of
phase transitions and properties of these
models. I will discuss the condensation regime
where these model undergo a one step replica symmetry
breaking transition.
Joint work with Nike Sun and Yumeng Zhang
- Joel Lebowitz
Fluctuations, Large Deviations and Rigidity in
Superhomogeneous Systems
Superhomogeneous (a.k.a. hyperuniform) particle
systems are point processes on R^d (or Z^d) that are
translation invariant (or periodic) and for which the
variance of the number of particles in a region V
grows slower than the volume of V. Examples include
Coulomb systems, determinantal processes with
projection kernels and certain perturbed lattice
models. I will first review some old work on
superhomogeneous systems and and then describe some
new work (with Subhro Ghosh) providing sufficient
conditions (involving decay of pair correlations) for
number rigidity in such systems in dimension d=1,2. A
particle system is said to exhibit number rigidity if
the probability distribution of the number of
particles in a bounded region R, conditioned on the
particle configuration in R^c, is concentrated on a
single integer N. All known (to us) examples in which
number rigidity has been established in d=1,2 satisfy
our conditions, and we conjecture that
superhomogeneity is also a necessary condition for
number rigidity in all dimensions d. On the other
hand, it follows from the work of Peres and Sly on
perturbed lattice systems that in d>2 there are no
such sufficiency conditions involving the decay of
correlations.
- Guillaume Barraquand
Random walks in Beta random environment.
We consider a model of random walks in space-time
random environment, with Beta-distributed transition
probabilities. This model is exactly solvable, in the
sense that the law of the (finite time) position of
the walker can be completely characterized by Fredholm
determinantal formulas. This enables to prove a limit
theorem towards the Tracy-Widom distribution for the
second order corrections to the large deviation
principle satisfied by the walker, thus extending the
scope of KPZ universality to RWRE. We will also
present a few similar results about degenerations of
the model: a first passage percolation model which is
the "zero-temperature" limit, and a certain diffusive
limit which leads to well-studied stochastic flows.
(Joint work with Ivan Corwin)
- Hao Shen
Regularity structure theory and its applications
I will review the basic ideas of the regularity
structure theory recently developed by Martin Hairer,
as well as its applications to stochastic PDE
problems. I will then discuss my joint works with
Hairer on well-posedness of the sine-Gordon equation,
and central limit theorems for stochastic PDEs.
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