Prob/Stat Seminar

This talk will review some of the main ideas that have appeared in the statistical literature in connection with the problem of estimating the dimension of the manifold where a data set lives, with an emphasis on the methods which are applicable in a local way (in neighborhoods of data points). Some recent developments in this area will be discussed, as well.

First-Passage Percolation (FPP) is a model first proposed by Hammersley and Welsh in 1965 to model fluid flow through a random medium. Over the past 50 years, FPP has drawn wide attention in biology, statistical physics, and mathematics. In this talk, I will first present a result about the asymptotic behavior of the time constant of the FPP model in high-dimensions. A recent progress on the phase-transition of the Chase-Escape model (an extension to the classic one-type FPP model to two types of particles) will be discussed, along with many interesting simulations and conjectures.

We study a variety of random walks on sub-Riemannian manifolds and their diffusion limits, which give, via their infinitesimal generators, second-order operators on the manifolds. A primary motivation is the lack of a canonical Laplacian in sub-Riemannian geometry, and thus we are particularly interested in the relationship between the limiting operators, the geodesic structure, and operators which can be obtained as divergences with respect to various choices of volume. This work is joint with Ugo Boscain (CNRS), Luca Rizzi (CNRS), and Andrei Agrachev (SISSA).

Stochastic processes on manifolds provide an important bridge between probability and geometry. The properties of Brownian motion on a Riemannian manifold depend strongly on properties of the geometry, such as its curvature. In this talk, I'll focus on Lie groups, which are manifolds with a lot of symmetry, and discuss how the weaker property of volume doubling can provide similar probabilistic results to what can be learned from curvature bounds.

On the 3-dimensional Lie group SU(2), we have proved that volume doubling can be uniformly controlled over all left-invariant metrics. I'll briefly discuss this geometric result, and what it tells us about doing probability on SU(2).

This is joint work with Maria Gordina and Laurent Saloff-Coste.

In this talk we study small time asymptotic of the heat content for a smoothly bounded domain with non-characteristic boundary in the Heisenberg group, which captures geometric information of the of the boundary including perimeter and the total horizontal mean curvature of the boundary of the domain.

We use probabilistic method by studying the escaping probability of the horizontal Brownian motion process that is canonically associated to the sub-Riemannian structure of the Heisenberg group. This is a joint work with J. Tyson.

We employ stabilization methods in the context of Malliavin-Stein theory to establish rates of multivariate normal convergence for a large class of vectors $$(H_s^{(1)},...,H_s^{(m)}), \ s \geq 1,$$ of marked Poisson point processes in Euclidean space, as the intensity parameter \(s \to \infty\). The rates are in terms of the \(d_2\) and \(d_3\) distances, a generalized multivariate Kolmogorov distance, as well as in terms of the convex distance defined in terms of indicators of convex sets. In general the rates are unimprovable. We the general results to deduce presumably optimal rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions.

Mean field spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean field models include the Sherrington-Kirkpatrick (SK) model, the (Ising) mixed p-spin model and the spherical mixed p-spin model. Starting in 1979, the physicist Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times at low temperature. In this talk, we will show that Parisi's prediction holds at zero temperature for the more general mixed p-spin model. On the other hand, we will show that there exist two-step RSB spherical mixed p-spin glass models at zero temperature, which are the first natural examples beyond the replica symmetric, one-step RSB and Full-step RSB phases.

This talk is based on joint works with Antonio Auffinger (Northwestern University) and Wei-Kuo Chen (University of Minnesota).

This talk will focus on selected models of random tessellations and the investigation of their geometrical characteristics. Tessellations arise naturally in many contexts. Examples include crystals, cellular structures, and communication networks. A random tessellation can be also viewed as a special case of a particle process, where some further conditions on the structure of particles (cells) are imposed. Usually, we are able to observe a realization of the tessellation only in some bounded window \(W\). Let \(\mathcal{X}\) be a random tessellation. One option is to choose a suitable functional \(H( \mathcal{X})\) and observe its values as the observation window increases. The natural choice is a functional of the the form of sum of local contributions $$H(\mathcal{X})= \sum\limits_{K \in \mathcal{X}} \xi(K,\mathcal{X}) \textbf{1}\{K \subset W\},$$ where \(\xi\) is called the score function (chosen so that the sum makes sense). This statistic may disregard the edge effects caused by observing a realization in a bounded window. One possibility for how to treat the edge effects is the minus sampling, i.e. we upgrade the summands by the following weights, $$H'(\mathcal{X}) = \sum\limits_{K \in \mathcal{X}} \frac{|W|}{|W\ominus K|}\, \xi(K,\mathcal{X}) \textbf{1}\{K \subset W\},$$ where \(W \ominus K = \{x \in \mathbb{R}^d: K+x \subset W\}\) and \(|A|\) stands for the Lebesgue measure of \(A\). We call this choice the Horvitz-Thompson statistic. It can be shown that \(H'(\mathcal{X})\) is an unbiased estimator of the expectation of \(\xi\) at the typical cell. Note that we are dealing with a sum of mutually dependent terms. To obtain a central limit theorem for \(H'(\mathcal{X})\), one has to use methods based on controlling the range of dependence. It will be shown how to use the stabilization method for this type of statistic.

Let B be the set of rooted trees that contain an infinite binary subtree starting at the root. This set satisfies the metaproperty of containing a tree if and only if it contains at least two of its root child subtrees. Suppose we wish to know the probability that a Galton–Watson tree falls in B. The metaproperty forces this probability to satisfy a fixed-point equation, which can have multiple solutions. One of these is the probability we seek, but what is the meaning of the other solutions? In particular, are they probabilities of the Galton–Watson tree falling into some other set satisfying the same metaproperty? We create a framework that lets us answer all questions of this sort. Our proofs use spine decompositions of Galton–Watson trees and the analysis of Boolean functions. Joint work with Moumanti Podder and Fiona Skerman.