BGU Probability and Ergodic Theory (PET) seminar Ical Atom

We will consider a class of groups defined by their action on Cantor space and use the invariant of finiteness properties to find among these groups an infinite family of quasi-isometry classes of finitely presented simple groups.

This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

We introduce the notion of stationary actions in the context of C-algebras, and prove a new characterization of C-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C-simplicity and the unique trace property, and provides a framework in which C-simplicity and random walks interact. Joint work with Mehrdad Kalantar.

We apply the Tur\´an sieve and the simple sieve developed by Ram Murty and Yu-Ru Liu to study problems in random graph theory. More specifically, we obtain bounds on the probability of a graph having diameter 2 (or diameter 3 in the case of bipartite graphs). An interesting feature revealed in these results is that the Tur´an sieve and the simple sieve “almost completely” complement to each other. This is joint work with Yu-Ru Liu.

The talk discusses a convexity structure on metric spaces which we call sheltered sets. This structure arises in the study of the dynamics of the maximum cellular automaton over the binary alphabet on finitely generated groups. I will discuss relations to horoballs and dead ends in groups and present many open questions. This is work in progress with Tom Meyerovitch.

We will present a model for construction of random fractals which is called fractal percolation. The main result that will be presented in this talk states that a typical fractal percolation set E intersects every set which is winning for a certain game that is called the “hyperplane absolute game”, and the intersection has the same Hausdorff dimension as E. An example of such a winning set is the set of badly approximable vectors in dimension d. In order to prove this theorem one may show that a typical fractal percolation set E contains a sequence of Ahlfors-regular subsets with dimensions approaching the dimension of E, where all the subsets in this sequence are also “hyperplane diffuse”, which means that they are not concentrated around affine hyperplanes when viewed in small enough scales. If time permits, we will sketch the proof of this theorem and present a generalization to a more general model for random construction of fractals which is given by projecting Galton-Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane.

The concept of Benjamini-Schramm convergence can be extended to Riemannian manifolds. In this setup a question frequently studied is whether topological invariants that can be expressed as integers are continuous when normalized by the volume. An example of such an invariant is the Euler characteristic, that also exists for graphs. A vast generalization of the Euler characteristic for Riemannian manifolds are characteristic numbers. I will speak on my results showing continuity of normalized characteristic numbers on a suitable class of random Riemannian manifolds defined by a lower Ricci curvature and injectivity radius bound.

A circle packing is a collection of circles in the plane with disjoint interiors. An accumulation point of the circle packing is a point with infinitely many circles in any neighborhood of it. A site percolation with parameter p on the circle packing means retaining each circle with probability p and deleting it with probability 1-p, independently between circles. We will explain the proof of the following result: There exists p>0 satisfying that for any circle packing with finitely many accumulation points, after a site percolation with parameter p there is no infinite connected component of retained circles, almost surely. This implies, in particular, that the site percolation threshold of any planar recurrent graph is at least p. It is conjectured that the same should hold with p=1/2. The result gives a partial answer to a question of Benjamini, who conjectured that square packings of the unit square admit long crossings after site percolation with parameter p=1/2 and asked also about other values of p. Time permitting, we will discuss an application of the result to the existence of macroscopic loops in the loop O(n) model on the hexagonal lattice. Portions joint with Nick Crawford, Alexandar Glazman and Matan Harel.

Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $\Omega_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on $\Omega_N$ which gives to each $n\in\Omega_N$ a probability proportional to $\frac{1}{n}$. This measure is in fact the distribution of the random integer $I_N\in\Omega_N$ defined by $I_N=\prod_{j=1}^Np_j^{X_{p_j}}$, where $\{X_{p_j}\}_{j=1}^\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\frac{1}{p_j})$. We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the Dickman distribution. As a corollary, we recover a classical result from classical multiplicative number theory—Mertens’ formula, which states that $\sum_{n\in\Omega_N}\frac{1}{n}\sim e^\gamma\log N$ as $N\to\infty$.

Let $D_{\text{nat}}(A)$ denote the natural density of $A\subset\mathbb{N}$, if it exists, and let $D_{\text{log-indep}}(A)=\lim_{N\to\infty}P_N(A\cap\Omega_N)$ denote the density of $A$ arising from $\{P_N\}_{N=1}^\infty$, if it exists. We show that the two densities coincide on a natural algebra of subsets of $\mathbb{N}$. We also show that they do not agree on the sets of $n^\frac{1}{s}$- smooth numbers $\{n\in\mathbb{N}: p^+(n)\le n^\frac{1}{s}\}$, $s>1$, where $p^+(n)$ is the largest prime divisor of $n$. This last consideration concerns distributions involving the Dickman function. We also consider the sets of $n^\frac{1}{s}$- rough numbers ${n\in\mathbb{N}:p^-(n)\ge n^{\frac{1}{s}}}$, $s>1$, where $p^-(n)$ is the smallest prime divisor of $n$. We show that the probabilities of these sets, under the uniform distribution on $[N]={1,\ldots, N}$ and under the $P_N$-distribution on $\Omega_N$, have the same asymptotic decay profile as functions of $s$, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.

Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

Krieger’s generator theorem shows that any free invertible ergodic measure preserving action $(Y,\mu, S)$ can be modelled by $A^Z$ (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is $A^Z$) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which $Z^d$-dynamical systems are universal. These conditions are general enough to prove that

1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet) 2) Proper colourings of the $Z^d$ lattice with more than two colours and the domino tilings of the $Z^2$ lattice (answering a question by Şahin and Robinson) are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.