BGU Probability and Ergodic Theory (PET) seminar Ical Atom

Delone sets are generalizations of point lattices: unformly discrete point sets with no large holes. In 1997 Gromov asked whether any Delone set in the Euclidean plane is bilipschitz equivalent to the integer lattice $Z^2$. A simpler but stronger condition than bilipschitz equivalence is bounded distance equivalence. So it is natural to ask which Delone sets in $R^d$ are bounded distance equivalent to (some scaled copy of) $Z^d$. This talk gives a gentle introduction to the problem and presents recent results in this context, mostly for cut-and-project sets on the line. In particular we show a connection between bouded remainder sets and cut-and-project sets that are bounded distance equivalent to some lattice.

An inhomogeneous Markov chain $X_n$ is a Markov chain whose state spaces and transition kernels change in time. A “local limit theorem” is an asymptotic formula for probabilities of the form

$Prob[S_N-z_N\in (a,b)]$, $S_N=f_1(X_1,X_2)+....+f_N(X_N,X_{N+1})$

in the limit $N\to\infty$. Here $z_N$ is a “suitable” sequence of numbers. I will describe general sufficient conditions for such results.

If time allows, I will explain why such results are needed for the study of certain problems related to irrational rotations.

This is joint work with Dmitry Dolgopyat.

Reflected diffusions (RDs) constrained to remain in convex polyhedral domains arise in a variety of contexts, including as heavy traffic limits of queueing networks and in the study of rank-based interacting particle models. Pathwise derivatives of an RD with respect to its defining parameters is of interest from both theoretical and applied perspectives. In this talk I will characterize pathwise derivatives of an RD in terms of solutions to a linear constrained stochastic differential equation that can be viewed as a linearization of the constrained stochastic differential equation the RD satisfies. The proofs of these results involve a careful analysis of sample path properties of RDs, as well as geometric properties of the convex polyhedral domain and the associated directions of reflection along its boundary.

This is joint work with Kavita Ramanan.

Let $X$ denote the number of triangles in the random graph $G(n,p)$. The problem of determining the asymptotic of the rate of the lower tail of $X$, that is, the function $f_c(n,p) = log Pr(X ≤ c E[X])$ for a given $c ∈ [0,1)$, has attracted considerable attention of both the combinatorics and the probability communities. We shall present a proof of the fact that whenever $p >> n^{-1/2}$, then $f_c(n,p)$ can be expressed as a solution to a natural combinatorial optimisation problem that generalises Mantel’s / Turan’s theorem. This is joint work with Gady Kozma.

The theory of mathematical quasicrystals essentially goes back to work of Meyer in the 70’s, who investigated aperiodic point sets in Euclidean space. Shechtman’s discovery of physical quasicrystals (1982, Nobel prize for Chemistry 2011) via diffraction experiments triggered a boom of the mathematical analysis of the arising scatter patterns. Recent years have seen some progress in understanding the geometry, Fourier theory and dynamics of well-scattered, aperiodic point sets in non-commutative groups. We explain some of those developments from the viewpoint of approximation of certain key quantities arising from the underlying group actions via a notion of convergence of dynamical systems. One particular focus in this context will be on sufficient criteria to ensure unique ergodicity of the dynamical system associated with a point set.

Based on joint projects with Siegfried Beckus and Michael Björklund/Tobias Hartnick.

Warped cones are families of metric spaces that can be associated with actions of discrete groups on compact metric spaces. They were first introduced by John Roe as means of producing interesting examples of metric spaces (in the context of the coarse Baum-Connes conjecture), and have since evolved as it turned out that they could be used to construct families of expander graphs and that they were good candidates for a definition of a `coarse geometric’ invariant of actions. In this talk I will introduce the warped cone construction and explain how to use it to obtain expanders. I will then indicate some rigidity results that hold in this settings.

Let T be a bounded linear operator on a Banach space X; the elements of (I − T)X are called coboundaries. For two commuting operators T and S, elements of (I − T)X ∩ (I − S)X are called joint coboundaries, and those of (I − T)(I − S)X are double coboundaries. By commutativity, double coboundaries are joint ones. Are there any other? Let θ and τ be commuting invertible measure preserving transformations of (Ω, Σ, m), with corresponding unitary operators induced on L2(m). We prove the existence of a joint coboundary g ∈ (I − U)L2 ∩ (I − V )L2 which is not in (I − U)(I − V )L2. For the proof, let E be the spectral measure on T 2 obtained by Stone’s spectral theorem. Joint and double coboundaries are characterized using E, and properties of the maximal spectral type of E, together with a result of Foia³ on multiplicative spectral measures acting on L2, are used to show the existence of the required function.

We will consider some basic optimization problems and how they relate to optics. We then define an index of refraction to any given distribution. We conjecture an estimate for the index and explain how its related to some natural operations research questions. We also consider lenses and ask questions about the probabilistic behavior of discrete geodesics in a lens setting.

A problem, arising naturally in the context of the coupon collector’s problem, is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS.:8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with the same parameters. We also deal with the probability of each of the variables being the maximal one.

The calculations lead to expressions involving Hurwitz’s zeta function at certain special points. We find here explicitly the values of the function at these points.