Research

 Research Interests:

My research is supported by NSF-DMS 2246727.

Publications

14- A central limit theorem for random walks on horospherical products of Gromov hyperbolic spaces]{A central limit theorem for random walks on horospherical products of Gromov hyperbolic spaces, submitted.

 (with A. Bhamanian, I. Gekhtman, K. Mallahi-Karai)

Abstract:

We develop a metric on the horospherical product of a finite number of proper Gromov hyperbolic spaces and study the asymptotic behavior of random walks with respect to this metric.  Our setting includes horospherical products of a finite number of trees, lamplighter groups over $\ZZ$, Diestel-Leader graphs, and Sol groups. Under finite second moment and non-zero drift conditions, we establish a central limit theorem for the displacement of a random walk on affine groups of these metric spaces. Along the way, we prove some geometric properties of these spaces.

13- The Poisson boundary of hyperbolic groups without moment conditions, arxiv.

(with K. Chawla, J. Frisch,  G. Tiozzo)

Abstract:

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.

12- Poisson representation and Furstenberg entropy of hypergroups, Journal of Functional Analysis (2023) Volume 285, Issue 9 (with K. Mallahi-Karai)

Link to the paper.  https://doi.org/10.1016/j.jfa.2023.110090

Abstract:

We extend the theory of Poisson boundary, tail boundary and the associated entropy theory of groups to the class of discrete hypergroups. We establish a zero-entropy criterion for the Liouville property of random walks on discrete hypergroups and provide various other characterizations for it. Finally, we will solve the identification problem for the Poisson boundary of finite range random walks on permutation hypergroups associated with affine groups of homogenous trees. As a byproduct, we obtain the first examples of random walks on hypergroups that have a countable infinite Poisson boundary.

11- Strong  Shannon- McMillan-Breiman's Theorem for locally compact groups, Canadian Mathematical Bulletin (to appear) Link to the paper.

DOI: https://doi.org/10.4153/S0008439523000358

(with M. Ngyuen*). 

*: M. Ngyuen was an undergraduate student at the College of Charleston.

Abstract:

We prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we partially answer Derriennic’s question [Der80]. More precisely, we show that the almost sure and L1 convergences of Shannon- McMillan-Breiman’s theorem hold for compactly supported random walks on compactly generated locally compact groups with subexponential growths.

10-Shannon's Theorem for locally compact groups, Annals of Probability (2022), 50 (1): 61-89

(with G. Tiozzo). 

Abstract:

 We consider random walks on locally compact groups, extending the geometric criteria for the identification of their Poisson boundary previously known for discrete groups. First, we prove a version of the Shannon- McMillan-Breiman theorem, which we then use to generalize Kaimanovich’s ray approximation and strip approximation criteria. We give several applications to identify the Poisson boundary of locally compact groups which act by isometries on nonpositively curved spaces, as well as on Diestel-Leader graphs and horocylic products.

9- Compactifications of horospheric products, L’enseignement Mathématique (2022), 

(with K. Mallahi-Karai)

Abstract:

We define and study a new compactification, called the height compactification of the horospheric product of two infinite trees. We will provide a complete description of this compactification. In particular, we show that this compactification is isomorphic to the Busemann compactification when all the vertices of both trees have degrees at least three, which also leads to a precise description of the Busemann functions in terms of the points in the geometric compactification of each tree.

8-On Transformations of Markov chains and Poisson boundaries, Transactions of the AMS, 373 (2020), 2207-2227 

https://doi.org/10.1090/tran/7975

(with I. Ben-Ari).

Abstract:

A discrete-time Markov chain can be transformed into a new Markov chain by looking at its states along iterations of an almost surely finite stopping time. By the optional stopping theorem, any bounded har- monic function with respect to the transition function of the original chain is harmonic with respect to the transition function of the transformed chain. The reverse inclusion is in general not true. Our main result provides a sufficient condition on the stopping time which guarantees that the space of bounded harmonic functions for the transformed chain embeds in the space of bounded harmonic sequences for the original chain. We also obtain a similar result on positive unbounded harmonic functions, under some additional conditions.

7-Random walks of infinite moment on free semigroups,  Probability Theory and Related Fields (2019) 175: 1099-1122

DOI: 10.1007/s00440-019-00911-7

(with G. Tiozzo).

Abstract:

We consider random walks on finitely or countably generated free semigroups, and identify their Poisson boundaries for classes of measures which fail to meet the classical entropy criteria, namely measures with infinite entropy or infinite logarithmic moment.


6-Positive Harmonic functions of transformed random walks, Potential Analysis 51,  563–578 (2019)

https://doi.org/10.1007/s11118-018-9724-4

(with K. Mallahi-Karai)

Abstract:

In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a discrete group under the change of probability measure by a randomized stopping time. We show that this space remains unchanged after applying a bounded randomized stopping time.

5-Asymptotic entropy of transformed random walks,  Ergodic Theory and Dynamical Systems,  Vol 37, October 2017, 1480-1491.

Abstract:

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk 38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys. 34(3) (2014), 837–853].

4-Amenability and trees,  London Mathematical Society, Lecture Notes Series 436 (2017), Groups, Graphs and Random Walks 

(with K. Mallahi-Karai).

Abstract:

We will give a criterion for the amenability of arbitrary locally finite trees. The criterion is based on the trimming operator which is defined on the space of trees. As an application, we obtain a necessary and sufficient condition for that amenability of Galton-Watson trees.

Theses

3-PhD. thesis, Transformed random walks,  University of Ottawa, Canada, 2015.

2-Master thesis, Amenability and trees,  Jacobs University of Bremen, Germany, 2010.

1-Master thesis, Some characterization of reflexive spaces, Sharif University of Technology, Tehran, Iran, 2007.

In preparation

-Convex Algebra and Harmonic measures

(with V. Kaimanovich).

Boundary preserving transformations of random walks

 (with V. Kaimanovich).

Abstract:

We describe a new construction of a family of measures on a group with the same Poisson boundary. Our approach is based on applying Markov stopping times to an extension of the original random walk. This method provides a new proof of Furstenberg Conjecture which states for any amenable group there exists a generating random walk with trivial Poisson boundary. 

-Entropy and Poisson boundary of groupoids 

(with V. Kaimanovich).