BGU Probability and Ergodic Theory (PET) seminar Ical Atom

We will give sufficient conditions for the bounded law of the iterated logarithms for strictly stationary random fields with summation on rectangles. The case of martingales differences with respect to the lexicographic order and the orthormartingales will be investigated, as well as martingale approximation.

Given a lattice $\Lambda$ and a (perhaps long) vector $v \in \Lambda$, we consider two geometric quantities: - the projection $\Delta$ of $\Lambda$ along the line through $v$ - the “lift functional” which encodes how one can recover $\Lambda$ from the projection $\Delta$ Fixing $\Lambda$ and taking some infinite sequences of vectors $v_n$, we identify the asymptotic distribution of these two quantities. For example, for a.e. line $L$, if $v_n$ is the sequence of $\epsilon$-approximants to $L$ then the sequence $\Delta(v_n)$ equidistributes according to Haar measure, and if $v'_n$ is the sequence of best approximants to $L$ then there is another measure which $\Delta(v'_n)$ equidistributes according to. The basic tool is a cross section for a diagonal flow on the space of lattices, and after some analysis of this cross section, the results follow from the Birkhoff pointwise ergodic theorem.

Joint work with Uri Shapira.

Recently Uri Gabor refuted an old conjecture stating that any finitary factor of an i.i.d process is finitarly isomorphic to an i.i.d process. Complementing Gabor’s result, in this talk, which is based on work in progress with Yinon Spinka, we will prove that any countable-valued process which is admits a finitary a coding by some i.i.d process furthermore admits an $\epsilon$-efficient finitary coding, for any positive $\epsilon$. Here an ‘’$\epsilon$-efficient coding’’ means that the entropy increase of the coding i.i.d process compared to the (mean) entropy of the coded process is at most $\epsilon$. For processes having finite entropy this in particular implies a finitary i.i.d coding by finite valued processes. As an application we give an affirmative answer to an old question about the existence of finite valued finitary coding of the critical Ising model, posed by van den Berg and Steif in their 1999 paper ‘‘On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields’’.

I’ll show the invalidity of finitary counterparts for three theorems in classification theory: The preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. This gives a negative answer to an old conjecture and to a recent open problem.

An observation by Jens Marklof shows that the primitive rational points of a fixed denominator along the periodic unipotent orbit of volume equal to the square of the denominator equidistribute inside a proper submanifold of the modular surface. This concentration as well as the equidistribution are intimately related to classical questions of number theoretic origin. We discuss the distribution of the primitive rational points along periodic orbits of intermediate size. In this case, we can show joint equidistribution with polynomial rate in the modular surface and in the torus. We also deduce simultaneous equidistribution of primitive rational points in the modular surface and of modular hyperbolas in the two-torus under certain congruence conditions. This is joint work with M. Einsiedler and N. Shah.

The usual weighted average of points $(z_0, ..., z_q)$ in the real vector space $R^n$, with weights $(w_0, ..., w_q)$, is translation invariant. Hence it can be seen as an average of points in a torsor Z over the Lie group $G = R^n$ (A $G$-torsor is a $G$-manifold with a simply transitive action.)

In this talk I will explain how this averaging process can be generalized to a torsor Z over a unipotent Lie group $G$. (In differential geometry, a unipotent group is a simply connected nilpotent Lie group. $R^n$ is an abelian unipotent group.)

I will explain how to construct the unipotent weighted average, and discuss its properties (functoriality, symmetry and simpliciality). If time permits, I will talk about torsors over a base manifold, and families of sections parametrized by simplices. I will indicate how I came about this idea, while working on a problem in deformation quantization.

Such an averaging process exists only for unipotent groups. For instance, it does not exist for a torus $G$ (an abelian Lie group that’s not simply connected). In algebraic geometry the unipotent averaging has arithmetic significance, but this is not visible in differential geometry.

Notes for the talk can be founds here: https://www.math.bgu.ac.il/~amyekut/lectures/average-diff-geom/abstract.html

A topological Markov shift is the set of two sided inifinite paths in a finite directed graph endowed with the product topology and with the left shift acting on this space. The automorphisms of the space are the shift commuting self-homeomorphisms. Wagoner realized the automorphism group of a topological Markov shift as the fundamental group of a certain CW complex. This construction has been crucial in many results regarding automorphisms and isomorphism in symbolic dynamics. We give a simplified construction of this complex, which also works in more general contexts, and sketch some applications.

A cut-and-project set is constructed by restricting a lattice $L$ in $(d+m)$-space to a domain bounded in the last m coordinates, and projecting these points to the the space spanned by its d-dimensional orthogonal complement. These point sets constitute an important example of so-called quasicrystals.

During the talk, we shall present and give some classification results of the moduli spaces of cut-and-project sets, which were introduced by Marklof-Strömbergsson. These are obtained by considering the orbit closure of the special linear group in $d$-space acting on the lattice $L$ inside the space of unimodular lattices of rank $d+m$. Theorems of Ratner imply that these are meaningful objects.

We then describe quantitative counting result for patches in generic cut-and-project sets. Patches are local configuration of point sets whose multitude reflects aperiodicity.

The count follows some old argument of Schmidt using moment bounds. These bounds are obtained by integrability properties of the Siegel transform, which in turn follow from reduction theory and a symmetrisation argument of Rogers. This argument is of independent interest, giving an alternative account to recent work of Kelmer-Yu (which is based on the theory of Eisenstein series) on counting points in generic symplectic lattices.

This is a joint endeavour with Yotam Smilansky and Barak Weiss.

Anosov flows are central objects in dynamics, generalizing the basic example of a geodesic flow over a Riemann surface.

In the talk we will introduce those flows and their dynamical behavior. Moreover, we show how the factorization method, pioneered by Eskin and Mirzakhani in their groundbreaking work about measure rigidity for the moduli space of translation surfaces, can be adapted to smooth ergodic theory and in particular towards the study of Anosov flows.

Using this adaption, we show that for a quantitatively non-integrable Anosov flow, every generalized u-Gibbs measure is absolutely continuous with respect to the whole unstable manifold.

The cutoff phenomenon of random walks on graphs is conjectured to be very common. However, it is unknown whether many natural examples of large graphs of fixed degree satisfy this phenomenon. It was recently shown by Lubetzky and Peres that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in optimal time. We show that the spectral condition can be replaced by a weaker spectral condition, based on the work of Sarnak and Xue in automorphic forms. This property is also equivalent to a geometrical path counting property, which can be verified in some cases. As an example, we show that the theorems hold for some families of Schreier graphs of the $SL_2(F_p)$ action on the projective line, for a finite field $F_p$. Based on joint work with Konstantin Golubev.

The Gaussian Entire Function (GEF) is the random Taylor series, whose coefficients are independent centered complex Gaussians such that the n-th coefficient has variance 1/n!. The zero set of the GEF is a random point process in the plane, which is invariant with respect to isometries. The topic of this talk is the zero distribution of the GEF conditioned on the event that no zero lies in a given (large) region.

If the hole is a disk of radius r, Ghosh and Nishry observed a striking feature. As r tends to infinity, the density of particles vanishes not only on the given hole, but also on an annulus beyond the (rescaled) hole — a forbidden region emerges. Here, we study this problem for general simply connected holes, and find a curious connection to quadrature domains and a seemingly novel type of free boundary problem.

This reports on joint work in progress with Alon Nishry.

The mixing time is a fundamental quantity measuring the rate of convergence of a Markov chain towards its stationary distribution. We will discuss the problem of estimating the mixing time from one single long trajectory of observations. The reversible setting was addressed using spectral methods by Hsu et al. (2015), who left the general case as an open problem. In the reversible setting, the analysis is greatly facilitated by the fact that the Markov operator is self-adjoint, and Weyl’s inequality allows for dimension-free perturbation analysis of the empirical eigenvalues. In the absence of reversibility, the existing perturbation analysis has a worst-case exponential dependence on the number of states. Furthermore, even if an eigenvalue perturbation analysis with better dependence on the number of states were available, in the non-reversible case the connection between the spectral gap and the mixing time is not nearly as straightforward as in the reversible case. We design a procedure, using spectral methods, that allows us to overcome the loss of self-adjointness and to recover a sample size with a polynomial dependence in some natural complexity parameters of the chain. Additionally, we will present an alternative estimation procedure that moves away from spectral methods entirely and is instead based on a generalized version of Dobrushin’s contraction. Joint work with Aryeh Kontorovich.

Estimating the Mixing Time of Ergodic Markov Chains Geoffrey Wolfer, Aryeh Kontorovich - COLT2019 http://proceedings.mlr.press/v99/wolfer19a.html
https://arxiv.org/abs/1902.01224

Mixing Time Estimation in Ergodic Markov Chains from a Single Trajectory with Contraction Methods Geoffrey Wolfer - ALT2020 https://arxiv.org/abs/1912.06845

The main result in this talk is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset S of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $\mu_1$ has the property that the random walk with initial distribution $\mu_1$ evolved by the action of S equidistributes very fast. The second measure $\mu_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.