| Critical values of smooth random fields and eigenvalues of random matrices
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Jonathan Taylor (Stanford)
We discuss the generic behaviour of the critical points/values of smooth Gaussian
random fields on smooth manifolds f:M \rightarrow \mathbb{R}, which we think of as
point processes on the parameter space of the field (the critical points) with
real-valued marks (the critical values).
For parameter spaces of a fixed dimension, these marked point processes can be combined with some
tools from differential topology to derive an accurate approximation to the supremum distribution
\mathbb{P}{ \sup_{x \in M} f(x) \geq u }
based on the geometry of the excursion sets
{ x \in M: f(x) \geq u },
specifically the expected Euler characteristic of the excursion sets.
In general, the accuracy of the above expression is poorly understood if we allow the
dimension of M to grow. In this work, we investigate some aspects of the accuracy in the high
dimensional setting, restricting attention to isotropic process on [0,1]^n with n growing. In
this situation, the "spectrum'' of critical values behave in some sense like the eigenvalues of a
large GOE (Gaussian Orthogonal Ensemble) matrix at the bulk and the edge.
We identify two separate regimes for the behaviour of the mean spectral measure of the
critical values of smooth isotropic Gaussian fields in high dimensions. Understanding the limiting
behaviour of the mean spectral measure depends on a characterization of the covariance functions of
isotropic processes in high dimensions.
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| The lower phase transition of the contact process (joint with M Aizenman)
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Paul Jung (Cornell)
Consider the contact process on any transitive graph with bounded
degree starting from one particle. Using a variational derivative form of
Russo's formula, we show that the expected
number of particles of the process decays exponentially in time
for all infection rates \lambda that are below the lower critical
value \lambda_s. Along the way we obtain certain critical exponent
bounds. Some of these results have previously been proven on
\mathbb{Z}^d in a well-known paper
of Bezuidenhout and Grimmett.
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| The critical random graph, with martingales
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Asaf Nachmias (Berkeley)
The random graph model, G(n,p), is obtained from the complete
graph on n vertices by independently retaining each edge with probability
p and erasing it with probability 1-p.
This model was introduced by Erdos and Renyi, who discovered that for
p=c/n, the largest connected component of G(n,p) undergoes a "double jump"
as c grows; its size is of order log(n) for c<1, of order n^{2/3} for c=1,
and linear for c>1. A complete proof for the case c=1 was only given much
later (see Luczak, Pittel and Wierman 1994, and Aldous 1997), however all
previous proofs of this fact are quite involved.
We present simple proofs of these facts. Our methods also yield a simple
proof for a Theorem of Bollobas concerning the supercritical phase, and
can be used to analyze other models, such as critical percolation on
random regular graphs and on regular expanders.
Joint work with Yuval Peres.
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| Optimal flow through the disordered lattice
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David Aldous (Berkeley)
After general remarks on the topic of flows through random networks,
I will outline a proof of the following result.
Consider routing traffic
on the N x N torus,
simultaneously between all source-destination pairs,
to minimize the cost
\sum_e c(e)f^2(e),
where f(e) is the volume of flow across edge e and
the c(e) form an i.i.d. random environment.
One can prove existence of a rescaled N \to \infty limit constant
for minimum cost, by comparison with an appropriate analogous
problem about minimum-cost flows across a M x M
subsquare of the lattice.
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| Giambelli compatible point processes
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Evgeny Strahov (Cal Tech)
We distinguish a class of random point processes which we call
Giambelli compatible point processes. Our definition was partly
inspired by determinantal identities for averages of products and
ratios of characteristic polynomials for random matrices.
It is closely related to the
classical Giambelli formula for Schur symmetric functions.
We show that orthogonal polynomial ensembles, z-measures on
partitions, and spectral measures of characters of generalized
regular representations of the infinite symmetric group generate
Giambelli compatible point processes. In particular, we prove
determinantal identities for averages of analogs of characteristic
polynomials for partitions.
Our approach provides a direct derivation of determinantal
formulas for correlation functions.
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| Random times, filtrations, and path decompositions
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Ashkan Nikeghbali (AIM)
In this talk, using techniques from enlargements of filtrations,
we introduce a new family of random times which are natural
generalizations of stopping times, and then give different
generalizations of the celebrated Willimas' path decomposition
for the standard Brownian Motion.
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| Beta-coalescents and continuous random trees
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Nathanael Berestycki (University of British Columbia)
Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999).
These processes describe the evolution of particles that undergo stochastic
coagulation in such a way that several blocks can merge at the same time to
form a single block. In the case where the measure Lambda has the
Beta(2-a,a) distribution, Birkner et al. recently used the Donnelly-Kurtz
lookdown construction to prove that Beta-coalescents can be obtained as the
time-changed genealogies of a continuous-state branching process with stable
branching mechanism. Here we use this result to prove that Beta-coalescents
can be further embedded in continuous stable random trees, for which much is
known due to recent progress of Duquesne and Le Gall. This produces a number
of results concerning the small-time behavior of Beta-coalescents. Most
notably, we get an almost sure limit theorem for the number of blocks at
small times, for the rescaled sizes of the blocks, and give the multifractal
spectrum corresponding to the emergence of blocks with atypical size. Also,
we are able to find asymptotics for several quantities of interest to
biologists in the context of population genetics.
This is joint work with Julien Berestycki (Univ. Marseille) and Jason
Schweinsberg (UCSD).
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| Optimal use of communication resources
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Abraham Neyman (Hebrew University of Jerusalem)
Consider the following puzzle. Two players are offered the following
game by a Casino. The Casino produces along sequence of colors x1,x2,....
The n-th color is either Red or Black for odd n, and either Red or Black
or Green for even n. The Casino will disclose the entire sequence of
colors to player 1 before the betting start. Player 1 can not communicate
openly the content of the sequence to player 2. The only mean for player 1
to communicate information is true the actual play of the gambling, whose
rules are now described. At round n, the players place their bet on the
forthcoming color in closed envelopes. Then, the color xn and the content
of both enevelopes are exposed, and the players score if all three colors
coincide. The question: what is the average score per round that the two
players can guarantee.
The seminar will describe how the solution of this puzzle and of additional
questions follow from the general theory developed in the paper
``The optimal use of communication resources''
(by Gossner, Hernandez, and Neyman).
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| Quenched invariance principle for random walk in random environments admitting a finite cycle representation
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Jean-Dominique Deuschel (TU Berlin)
We consider a class of random walks in a random
environment on Z^d admitting a finite cycle
representation, that is the corresponding
jump rates are labeled by finite oriented cycles with
ergodic weights, e.g. [K], [M]. The reversible random conductances model
with trivial two points cycles is a particular case, see [S]
thus our model extends to the non reversible situation.
Assuming uniform irreducibility, we prove a quenched
invariant principle for the rescaled process.
The annealed CLT result has been proved recently in the special
case of two-fold walks by Komorovski and Olla in [K].
We adapt the quenched proof of Sidoravicius and Sznitman, [S],
to the non reversible case using corrector,
the sector condition and the
heat kernels upper bounds for centered random walks by Mathieu, [M].
Joint work with Holger Koesters.
[K] Komorowski, T; Olla, S., A note on the central limit theorem for two-fold stochastic random walks
in a random environment. Preprint (2005).
[M] Mathieu, P., Carne-Varopoulos bounds for centered random walks. Ann. of Prob. (2006).
[S] Sidoravicius, V.; Sznitman, A., Quenched invariance principles for walks on clusters
of percolation or among random conductances. Probab. Theory Related Fields, 129, 219-244.
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| Some functional inequalities and their application to the isoperimetric problem
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Cyril Roberto (Marne-la-Vallée)
In the first part of the talk we introduce the well known Poincaré and logarithmic
Sobolev inequalities, for probability measure on the real line and more generally on
R^n. We give some of their properties and applications, about concentration of measure
phenomenum, contractivity of the underlying semi-group and isoperimetry....
On the real line, we shall see that the Poincaré inequality is closely related to the two-side
exponential measure, while the logarithmic Sobolev inequality is more about the Gaussian one.
In the second part, we introduce more general functional inequalities that allow us to deal with
general measure with convex potential between exponentail and Gaussian. In particular we derive
dimension free isoperimetric inequalities for these measures.
This is joint work with Franck Barthe (Toulouse) and Patrick Cattiaux (Nanterre and Polytechnique).
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| Continuum tree asymptotics of discrete fragmentations and
applications to phylogenetic models
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Matthias Winkel (Oxford)
At the beginning of this talk I will discuss simple discrete Markovian
models of successive fragmentation of masses of integer sizes and
fragmentations of the set {1,...n}. We express the transition
probabilities in the form of a splitting rule. The branching
structure of the fragmentations is naturally encoded in a tree. Motivating
examples are due to Aldous (beta-splitting models) and Ford (alpha
models). These authors also introduce and study a notion of sampling
consistency as n varies. We will provide an integral representation for
(exchangeable) sampling consistent splitting rules.
The main part of the talk will then turn to the continuous-time
continuous-mass analogues of these discrete fragmentation processes,
mainly due to Bertoin, and establish strong convergence results in the
Gromov-Hausdorff sense of rescaled discrete fragmentation trees to
certain continuum random trees called self-similar fragmentation trees -
these trees were studied before by Haas and Miermont, the notion of
convergence was developed for probabilistic applications by Evans, Pitman
and Winter.
As an application we obtain continuum random tree limits for Aldous's
beta-splitting models and Ford's alpha models confirming in a strong way
that the whole discrete tree grows at the same speed as the mean height of
a randomly chosen leaf - such quantities were studied by Aldous and Ford.
Aldous conjectured weak convergence of leaf-height functions for his
beta-splitting models, which we can now prove as a slight variation of our
first convergence result - a full convergence result was known previously
only for approximations of Aldous's Brownian continuum random tree.
This is joint work with Benedicte Haas (Paris Dauphine), Gregory Miermont (Orsay), and
Jim Pitman (Berkeley).
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| Estimating mixing times via the spectral profile
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Sharad Goel (Stanford)
Given 52 playing cards, how many shuffles does it take to approximately
randomize the deck? More generally, how long does it take a finite Markov
chain to get close to its stationary distribution? In this talk, I'll
introduce the spectral profile as a tool for proving upper and lower bounds
on convergence rates. This approach extends the commonly used spectral gap
method, and allows us to recover and refine previous conductance-based
estimates of mixing time. I will illustrate how the spectral profile
technique is applied in several models, including groups with moderate
growth, the fractal-like Viscek graphs, and the torus. This work is joint
with Ravi Montenegro and Prasad Tetali.
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| Digital snowflakes
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Janko Gravner (Davis)
We will discuss several mathematical models of
snow crystal growth. For a popular class of cellular automata
known as Packard's Snowflake, one can develop a fairly
complete rigorous theory, addressing limiting density, fractal shapes
and exact solvability. The bulk of this theory is limited to
the deterministic cases, altough something can be said about
random perturbations. The talk will also address
a more realistic mesoscopic snowflake model, which offers
some hope of at least empirical analysis. This talk will
be accessible to most undergraduates and is on joint work
with David Griffeath (Univ. of Wisconsin).
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| Tilings, groves, multi-set permutations and quantum
random walks: multivariate generating functions with
cone singularities
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Robin Pemantle (U Penn)
A number of problems in combinatorics and probability
may be encoded into a generating function, F, and
limit theorems extracted analytically. The extraction
depends on details of the function F. I will discuss
the case where F = G/H is a rational 3-variable function
and H(1+x,1+y,1+z) is a conic.
Why should you care about this case? It turns out that
this encompasses many cases of the "Arctic Circle"
phenomena. That is, the limits of the quantities
such as domino placement probabilities, edge probabilities,
and locations of a quantum random walker will obey
a law that spreads out over a linearly growing disk
with a prescribed renormalized limit on the disk. This
work is in progress and is joint with Yuliy Baryshnikov.
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| Multiple SLEs
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Julien Dubedat (Courant)
Schramm-Loewner Evolutions (SLEs) have proved a powerful tool to describe the scaling limit of a conformally invariant simple curve. In several instances (percolation, uniform spanning tree ...), one can define in a discrete setting several simple curves. We will discuss questions pertaining to the joint law of these curves in the scaling limit.
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| Uquity of synonymity: almost all large binary trees are not
uniquely identified by their spectra or their immanantal polynomials
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Erick Matsen (Harvard)
Phylogenetic tree shape statistics are numerical summaries of some aspect
of the shape of a phylogenetic tree. Up to this time, most of the work on
tree shape has been done using ad-hoc formulas which attempt to quantify
some visible feature of tree shape. In this talk I will present some joint
work with Steve Evans investigating a more mathematically natural approach
based on matrix representations of the tree. The matrix representations
we consider are the adjacency matrix, the Laplacian matrix (that is, the
infinitesimal generator of the natural random walk), and the matrix of
pairwise distances between leaves. We show for any of these choices of
matrix that the fraction of binary trees with a unique spectrum goes to
zero as the number of leaves goes to infinity.
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