The spectra of lamplighter groups and Cayley machines

Communications in Theoretical Physics, 2009

In this paper we define direct product of graphs and give a recipe for obtained probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph obtain by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determine probability of walk on complicated graphs. Using this method, we calculate the probability of continuous-time classical and quantum random walks on many of finite direct product cayley graphs ( complete cycle, complete K n , charter and n-cube). Also, we inquire that the classical state the stationary uniform distribution is reached as t −→ ∞ but for quantum state is not always satisfy.

2019

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

Mathematische Annalen

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 underlying dynamical system 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 show absence of eigenvalues and establish equality of the integrated density of states and the von-Neumann-Kesten-Serre trace. Along the way we give a careful study of combinatorial and dynamical features of the subshift associated to the non-primitive substitution. In particular, we show that this subshift is an almost everywhere one-to-one extension of its maximal equicontinuous factor.

Groups, Graphs and Random Walks

There is a recently discovered connection between the spectral theory of Schrödinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators. Contents Introduction 1. Subshifts and aperiodic order in one dimension 2. Schrödinger operators with aperiodic order 2.1. Constancy of the spectrum and the integrated density of states (IDS) 2.2. The spectrum as a set and the absolute continuity of spectral measures 2.3. Aperiodic order and discrete random Schrödinger operators 3. The substitution τ , its finite words Sub τ and its subshift (Ω τ , T) 3.1. The substitution τ and its subshift: basic features 3.2. The main ingredient for our further analysis: n-partition and n-decomposition 3.3. The maximal equicontinuous factor of the dynamical system (Ω τ , T) 3.4. Powers and the index (critical exponent) of Sub τ 3.5. The word complexity of Sub τ 3.6. Generating the fixed point η by an automaton 3.7. Replacing τ by a primitive substitution 4. Background on graphs and dynamical systems 5. Grigorchuk's group G, its Schreier graphs and the associated Laplacians 5.1. Grigorchuk's group G 5.2. The Schreier graphs of G and the dynamical system (X, G) 5.3. Laplacians associated to the Schreier graphs of G 6. The connection between (X, G) and (Ω τ , T) 7. Spectral theory of the M ξ for ξ ∈ ∂T 8. Integrated density of states and Kesten-von-Neumann-Serre spectral measure References

Journal of Geometric Analysis, 2000

We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry. 1. Introduction Let G be a finitely generated group. For any finite generating set S satisfying S = S-1 , consider the Cayley graph (G, S) with vertex set G and an edge from x to y if and only if y = xs for some s ~ S. Thus, edges are oriented but this is merely a convention since (x, y) is an edge if and only if (y, x) is an edge. We allow the identity element id to be in S in which case our graph has a loop at each vertex. Clearly the graph (G, S) is invariant under the left action of G. Denote by Ixl the distance from the neutral element id to x in the Cayley graph (G, S), that is, ]xl is the minimal number k of elements of S needed to write x as x = sis2 .. 9 sk, si ~ S. The volume growth function of (G, S) is defined by V(n) =#Ix ~ a : Ixl _< n]. This paper focuses on the probability of return after 2n steps of the simple random walk on (G, S). For a survey of this topic, see [36]. The simple random walk on (G, S) is the Markov process (Xi)~ c with values in G which evolves as follows: If the current state is x, the next state is a neighbor of x chosen uniformly at random. This implicitly defines a probability measure Ps on G r~ such that Ps (Xn = y/ Xo = x) = Iz (sn' (x-ly) where 1 Ixs(g) = ~-~ls(g) and/1 ~n) is the n-fold convolution power of/x. Following usual notation we will also write P~(.) = Ps('/Xo = x) for the law of the walk based on S and started at x e G. To avoid parity problems, we consider only the probability of return at even times and set 4)s(n) = P~ (XEn = id) =/x(s2n)(id) 9