Tatiana Nagnibeda

Random Walks on Trees with Finitely Many Cone Types

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Abstract.Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (Fk) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs.

Partition Functions of the Ising Model on Some Self-similar Schreier Graphs

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk’s group of intermediate growth; the iterated monodromy group of the complex polynomial z 2-1 known as the “Basilica group”; and the Hanoi Towers group H (3) closely related to the Sierpinski gasket.

Counting dimer coverings on self-similar Schreier graphs

We study partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions of finitely generated groups by automorphisms on rooted trees, and thus their edges are naturally labeled by the generators of the group. It is thus natural to consider weight functions on these graphs taking different values according to the labeling. We study in detail the well-known example of the Hanoi Towers group H ( 3 ) , closely related to the Sierpinski gasket.

Not Every Uniform Tree Covers Ramanujan Graphs

The notion of Ramanujan graph has been extended to not necessarily regular graphs by Y. Greenberg. We construct infinite trees with infinitely many finite quotients, none of which is Ramanujan. We give a sufficient condition for a finite graph to be covered by such a tree.

Spectra of Schreier graphs of Grigorchuk's group and Schroedinger operators with aperiodic order

We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk’s group acting on the boundary of the infinite binary tree. We establish a connection between the action of G on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.

Weighted Coxeter graphs and generalized geometric representations of Coxeter groups

We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.

Small spectral radius and percolation constants on non-amenable Cayley graphs

Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Gamma, does there exist a generating set S such that the Cayley graph (Gamma, S), without loops and multiple edges, has non-unique percolation, i.e., p(c)(Gamma, S) = 2, p >= 665. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group.

Lamplighter groups, de Brujin graphs, spider-web graphs and their spectra

We study the infinite family of spider-web graphs , , and , initiated in the 50s in the context of network theory. It was later shown in physical literature that these graphs have remarkable percolation and spectral properties. We provide a mathematical explanation of these properties by putting the spider-web graphs in the context of group theory and algebraic graph theory. Namely, we realize them as tensor products of the well-known de Bruijn graphs with cyclic graphs and show that these graphs are described by the action of the lamplighter group on the infinite binary tree. Our main result is the identification of the infinite limit of , as , with the Cayley graph of the lamplighter group which, in turn, is one of the famous Diestel–Leader graphs . As an application we compute the spectra of all spider-web graphs and show their convergence to the discrete spectral distribution associated with the Laplacian on the lamplighter group.

Universal groups of intermediate growth and their invariant random subgroups

We exhibit examples of groups of intermediate growth with