
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

201415
Toronto Probability Seminar (Sept.
8, 2014 to March 30, 2015)
held
at the Monday 2:10 p.m. in Stewart Library
For
questions, scheduling, or to be added to the
mailing list, contact the organizers at: probsem<@>math.toronto.ed


Organizers
Bálint
Virág , Janosch
Ortmann University of Toronto, Mathematics
and Statistics



Mondays 
Upcoming
Seminars in the Fields Institute, Stewart library

2:10pm
Monday March 16, 2015 
Tom Alberts (Utah)
Random Geometry in the Spectral Measure of the Circular Beta Ensemble
The Circular Beta Ensemble is a family of random unitary matrices
whose eigenvalue distribution plays an important role in statistical
physics. The spectral measure is a canonical way of describing the
unitary matrix that takes into account the full operator, not just
its eigenvalues. When the matrix is infinitely large (i.e. an operator
on some infinitedimensional Hilbert space) the spectral measure is
supported on a fractal set and has a rough geometry on all scales.
This talk will describe the analysis of these fractal properties.
Joint work in progress with Raoul Normand and Bálint Virág.

Past Talks 
2:10pm
Monday February 23, 2015 
Takis Konstantopoulos (Uppsala University)
Finite exchangeability
Whereas de Finetti's theorem characterizes exchangeable probability
measures on infinite product spaces under some topological assumptions,
analogous results for finite products have received less attention.
We give a short proof of the theorem stating that finitely exchangeable
product measures are "signed mixtures" of product measures
and then proceed in giving necessary and sufficient conditions for
extendibility of exchangeable probability measures.This is joint work
with Linglong Yuan.

2:10pm
Monday February 2, 2015

Vladimir Vinogradov (Ohio University, on leave at the Fields
Institute, University of Toronto and York University)
On equi/over/underdispersion and related properties of members
of some classes of probability distributions (slides)
The first part pertains to the class of PoissonPoisson distributions
for which an alternative name the "overdispersed Poisson"
was used during January 13, 2015 presentation by N. Altman:
http://www.fields.utoronto.ca/videoarchive/2015/01/3154118
We derive the closedform representation for the variance function
of this family. Since our representation involves the Lambert W function,
its combination with properties of this function enables one to evaluate
"deviations" from equidispersion. In particular, the index
of dispersion is locally one at the origin and grows with the logarithmic
speed at infinity. We employ this family to illustrate the author's
joint result on the general domain of attraction to Poisson laws,
which is parallel to the classical limit theorems on weak convergence
to stable distributions.
The second part concerns the extended family of zeromodified geometric
distributions whose representatives emerge in numerous stochastic
models. We characterize specific properties of the members of this
family via the value of an invariant of the Esscher transformation
(which is often called the exponential tilting), and obtain the variance
function for each such class in the closed form. We discover a new
family of equidispersed distributions which constitutes a proper subclass
of the extended family of zeromodified geometric probability laws.
We establish selfreciprocity in LetacMora sense for each family
of Esscher transforms comprised of its own zeromodified geometric
distributions, and present a unified formula for Shannon entropy of
all the representatives of the class of zeromodified geometric distributions.

2:10pm
Monday January 26, 2015 
Daniel Roy (Joint work with Nate Ackerman, Jeremy
Avigad, Cameron Freer and Jason Rute)
Conditional Independence, Computability, and Measurability
A subcommunity of machine learningworking in an area called
probabilistic programmingare now using sampling programs as specifications
for complex probability distributions over large collections of random
variables, and writing very general algorithms for computing conditional
distributions. The efficiency of these algorithms generally relies upon
an abundance of conditional independences. In this work, we look at
the problem of representing conditional independencies that hold among
these random variables. In particular, we look at the setting of exchangeable
sequences and arrays of random variables. The study of the computability
of representation theorems due to de Finetti, Aldous, and Hoover reveals
that representing conditional independence can come at a steep cost,
but also that a change of representationto one allowing a small probability
of errorallows us to represent conditional independences in these
random structures.

2:10pm
Monday Nov. 3, 2014 
Marcin Kotowski (Toronto).
Random Schroedinger operators and NovikovShubin invariants of groups
I will talk about random Schroedinger operators with random edge
weights and their expected spectral measures T near zero. We prove
that the measure exhibits a spike of the form C / (log eps)^2 (first
observed by Dyson), without assuming independence or any regularity
of edge weights. We then use the result to compute NovikovShubin
invariants for various groups, including lamplighter groups and lattices
in the Sol group. Joint work with Balint Virag.

2:10pm
Monday October 20, 2014 
Mustazee
Rahman (Toronto)
Phase Transitions in Random Cech complexes
It is well known that Bernoulli percolation on the dregular tree
has finite clusters so long as the density is at most 1/(d1). Now
consider a natural generalizing: an invariant percolation process
on the dregular tree that is a factor of an IID process such that
the factor map commutes with automorphisms of the tree. What is the
largest density of such a percolation if its clusters are finite?
A simple greedy algorithm provides a lower bound of (log d)/d for
large d. This bound also turns out to be asymptotically optimal in
d. We will explain this result and illustrate some ideas behind the
proof.

Tuesday
October 14,
11am
Room 210
Fields Institute

Omer Bobrowski (Duke)
Phase Transitions in Random Cech complexes
A simplicial complex is a collection of vertices, edges, triangles,
and simplexes of higher dimensions, and one can think of it as a generalization
of a graph. Given a random set of points P in a metric space and a
real number r > 0, one can create a simplicial complex by looking
at the balls of radius r around the points in P, and adding a kdimensional
face for every subset of k+1 balls that has a nonempty intersection.
This construction produces a random topological space known as the
Cech complex  C(P,r). We wish to study the homology of this space,
more specifically  its Betti numbers  the number of connected components
and 'holes' or 'cycles'.
In this talk we discuss the limiting behavior of the random Cech
complex as the number of points in P goes to infinity and the radius
r goes to zero. We show that the limiting behavior exhibits multiple
phase transitions at different levels, depending on the rate at which
the radius goes to zero. We present the different regimes and phase
transitions discovered so far, and observe the nicely ordered fashion
in which cycles of different dimensions appear and vanish. One interesting
consequence of this analysis is a sufficient condition for the random
Cech complex to successfully recover the homology of the support of
the distribution used to generate the data.
The talk will assume no prior knowledge in algebraic topology.

14:10 on Monday September 29, 2014
Stewart Library, Fields Institute

Michal Kotowski (Toronto)
I will talk about constructing a finitely generated group $G$ without
the Liouville property such that the return probability of a random
walk satisfies $p_{2n}(e,e) \gtrsim e^{n^{\gamma + o(1)}}$ for $\gamma
= 1/2$. This shows that the constant $1/2$ in a recent theorem by
Gournay, saying that return probability exponent less than $1/2$ implies
the Liouville property, cannot be improved.
The construction is based on permutational wreath products over treelike
Schreier graphs and the analysis of large deviations of inverted orbits
on such graphs. Joint work with Bálint Virág.

Monday, September
8,
22.50pm

François Huveneers (Dauphine).
Random walk driven by the simple exclusion process
I'll discuss a law of large numbers for a random walk in a onedimensional,
dynamical, binary random environment evolving as the simple exclusion
process. Both the quasistatic regime (slow evolution of the environment)
and the almost homogeneous regime (fast evolution) will be considered.
In some cases, the asymptotic velocity of the walker undergoes a transition,
i.e. flips signs, between these two regimes. After explaining the
main intuition beyond this phenomenon (trapping or not), I will introduce
the two principal technical tools used in the derivation of the result:
a renormalization procedure (to show that anomalous regions are irrelevant),
and a renewal structure (to derive the convergence to a limit). From
a joint work with F. Simenhaus (Paris Dauphine).

Wednesday,
September,
10,
4.104pm
Bahen Rm 6183

Bálint Virág
(Budapest/Toronto)
Operator limits of random matrices 
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