SCIENTIFIC PROGRAMS AND ACTIVITIES

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 infinite-dimensional 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.

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 Poisson-Poisson distributions for which an alternative name the "overdispersed Poisson" was used during January 13, 2015 presentation by N. Altman:
http://www.fields.utoronto.ca/video-archive/2015/01/315-4118

We derive the closed-form 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 zero-modified 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 zero-modified geometric probability laws. We establish self-reciprocity in Letac-Mora sense for each family of Esscher transforms comprised of its own zero-modified geometric distributions, and present a unified formula for Shannon entropy of all the representatives of the class of zero-modified geometric distributions.

Daniel Roy (Joint work with Nate Ackerman, Jeremy Avigad, Cameron Freer and Jason Rute)
Conditional Independence, Computability, and Measurability

A sub-community of machine learning---working in an area called probabilistic programming---are 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 representation---to one allowing a small probability of error---allows us to represent conditional independences in these random structures.

Marcin Kotowski (Toronto).
Random Schroedinger operators and Novikov-Shubin 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 Novikov-Shubin invariants for various groups, including lamplighter groups and lattices in the Sol group. Joint work with Balint Virag.

It is well known that Bernoulli percolation on the d-regular tree has finite clusters so long as the density is at most 1/(d-1). Now
consider a natural generalizing: an invariant percolation process on the d-regular 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 k-dimensional 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 tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs. Joint work with Bálint Virág.

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 one-dimensional, dynamical, binary random environment evolving as the simple exclusion process. Both the quasi-static 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).