Lamplighter Graph Research Articles

Stochastic approximation of lamplighter metrics

We observe that embeddings into random metrics can be fruitfully used to study the L 1 $L_1$ -embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L 1 $L_1$ -distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n $n$ -point metric space embeds bi-Lipschitzly into L 1 $L_1$ with distortion O ( log n ) $O(\log n)$ . In particular, for every finite group G $G$ the lamplighter group H = Z 2 ≀ G $H=\mathbb {Z}_2\wr G$ bi-Lipschitzly embeds into L 1 $L_1$ with distortion O ( log log | H | ) $O(\log \log |H|)$ . In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L 1 $L_1$ of the lamplighter group over the d $d$ -dimensional infinite lattice Z d $\mathbb {Z}^d$ can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d $d$ -dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

Random walks on the discrete affine group

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.

On the Bi-Lipschitz Geometry of Lamplighter Graphs

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into $\ell_1$. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterization in terms of a sequence of graphs with uniformly bounded degree of the notion of trivial Bourgain-Milman-Wolfson type for arbitrary metric spaces, similar to Ostrovskii's characterization previously obtained in \cite{ostrovskii:11}.

Cut-off for lamplighter chains on tori: dimension interpolation and Phase transition

Given a finite, connected graph $$\mathsf {G}$$ , the lamplighter chain on $$\mathsf {G}$$ is the lazy random walk $$X^\diamond $$ on the associated lamplighter graph $$\mathsf {G}^\diamond =\mathbb {Z}_2 \wr \mathsf {G}$$ . The mixing time of the lamplighter chain on the torus $$\mathbb {Z}_n^d$$ is known to have a cutoff at a time asymptotic to the cover time of $$\mathbb {Z}_n^d$$ if $$d=2$$ , and to half the cover time if $$d \ge 3$$ . We show that the mixing time of the lamplighter chain on $$\mathsf {G}_n(a)=\mathbb {Z}_n^2 \times \mathbb {Z}_{a \log n}$$ has a cutoff at $$\psi (a)$$ times the cover time of $$\mathsf {G}_n(a)$$ as $$n \rightarrow \infty $$ , where $$\psi $$ is an explicit weakly decreasing map from $$(0,\infty )$$ onto [1 / 2, 1). In particular, as $$a > 0$$ varies, the threshold continuously interpolates between the known thresholds for $$\mathbb {Z}_n^2$$ and $$\mathbb {Z}_n^3$$ . Perhaps surprisingly, we find a phase transition (non-smoothness of $$\psi $$ ) at the point $$a_*=\pi r_3 (1+\sqrt{2})$$ , where high dimensional behavior ( $$\psi (a)=1/2$$ for all $$a \ge a_*$$ ) commences. Here $$r_3$$ is the effective resistance from 0 to $$\infty $$ in $$\mathbb {Z}^3$$ .

Lamplighter Random Walks on Fractals

We consider on-diagonal heat kernel estimates and the laws of the iterated logarithms for a switchwalk-switch random walk on a lamplighter graph under the condition that the random walk on the underlying graph enjoys sub-Gaussian heat kernel estimates.

Harmonic functions with finite p-energy on lamplighter graphs are constant

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in ℓp (i.e. finite p-energy) for any p∈[1,∞[ except constants (and, equivalently, that their reduced ℓp cohomology is trivial in degree one). Similar arguments are then applied to many direct products of graphs to conclude the same (including all direct products of Cayley graphs). The proof relies on a theorem of Thomassen [16] on spanning lines in squares of graphs.

Uniform mixing time for random walk on lamplighter graphs

Soit $\mathcal{G} $ un graphe connexe fini et $X$ une marche aléatoire fainéante sur $\mathcal{G} $. La chaîne de l’allumeur de réverbères $X^{\diamond}$ associée à $X$ est la marche aléatoire sur le groupe produit $\mathcal{G} ^{\diamond}=\mathbf{Z} _{2}\wr \mathcal{G} $, le graphe dont les sites sont des paires $(\underline{f} ,x)$ où $f$ est un label des sites de $\mathcal{G} $ par des éléments de $\mathbf{Z} _{2}=\{0,1\}$ et $x$ est un site de $\mathcal{G} $. Il existe une arête entre $(\underline{f} ,x)$ et $(\underline{g} ,y)$ dans $\mathcal{G} ^{\diamond}$ si et seulement si $x$ est adjacent à $y$ dans $\mathcal{G} $ et $f_{z}=g_{z}$ pour tout $z\neq x,y$. A chaque pas, $X^{\diamond}$ se déplace d’une configuration $(\underline{f} ,x)$ en mettant à jour $x$ vers $y$ par la règle de translation de $X$ et ensuite en mettant à jour à la fois $f_{x}$ et $f_{y}$ selon la distribution uniforme sur $\mathbf{Z} _{2}$; $f_{z}$ pour $z\neq x,y$ restant inchangé. Nous prouvons des bornes supérieures et inférieures équivalentes sur le temps de mélange uniforme de $X^{\diamond}$ sous des hypothèses faibles sur $\mathcal{G} $. En particulier quand $\mathcal{G} $ est l’hypercube $\mathbf{Z} _{2}^{d}$, nous montrons que le temps de mélange uniforme de $X^{\diamond}$ est $\varTheta(d2^{d})$. Plus généralement, nous montrons que quand $\mathcal{G} $ est le tore $\mathbf{Z} _{n}^{d}$ avec $d\geq 3$, le temps de mélange uniforme de $X^{\diamond}$ est $\varTheta(dn^{d})$ uniformément en $n$ et $d$. Un ingrédient crucial de notre preuve est une estimation de concentration pour le temps local d’une marche aléatoire dans un sous ensemble de sites.

FINITE VORONOI DECOMPOSITIONS OF INFINITE VERTEX TRANSITIVE GRAPHS

In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of the number of balls of radius r-1 required to cover a sphere of radius r. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of s(G) is. We show that all vertex-transitive graphs with polynomial growth have finite s(G); vertex-transitive graphs with infinitely many ends have infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has finite s(G); and the lamplighter graph LL(Z2), which is Liouville, has infinite s(G).

The logarithmic Sobolev constant of the lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.

Uniformity of the uncovered set of random walk and cutoff for lamplighter chains

We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, with elements of ${0,1}$ given by i.i.d. fair coin flips on the range $\mathcal {R}$ of a random walk $X$ run until time $T$ and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold $T=1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoff with threshold $1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. We give a general criterion under which both of these results hold; other examples for which this applies include bounded degree expander families, the intersection of an infinite supercritical percolation cluster with an increasing family of balls, the hypercube and the Caley graph of the symmetric group generated by transpositions. The proof also yields precise asymptotics for the decay of correlation in the uncovered set.

Lamplighter graphs do not admit harmonic functions of finite energy

We prove that a lamplighter graph of a locally finite graph over a finite graph does not admit a non-constant harmonic function of finite Dirichlet energy.

A NOTE ON THE NORMS OF TRANSITION OPERATORS ON LAMPLIGHTER GRAPHS AND GROUPS

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓp-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ2-spectral radius of symmetric random walks on such groups.

Mixing Times for Random Walks on Finite Lamplighter Groups

Given a finite graph $G$, a vertex of the lamplighter graph $G^\diamondsuit=\mathbb {Z}_2 \wr G$ consists of a zero-one labeling of the vertices of $G$, and a marked vertex of $G$. For transitive $G$ we show that, up to constants, the relaxation time for simple random walk in $G^\diamondsuit$ is the maximal hitting time for simple random walk in $G$, while the mixing time in total variation on $G^\diamondsuit$ is the expected cover time on $G$. The mixing time in the uniform metric on $G^\diamondsuit$ admits a sharp threshold, and equals $|G|$ multiplied by the relaxation time on $G$, up to a factor of $\log |G|$. For $\mathbb {Z}_2 \wr \mathbb {Z}_n^2$, the lamplighter group over the discrete two dimensional torus, the relaxation time is of order $n^2 \log n$, the total variation mixing time is of order $n^2 \log^2 n$, and the uniform mixing time is of order $n^4$. For $\mathbb {Z}_2 \wr \mathbb {Z}_n^d$ when $d\geq 3$, the relaxation time is of order $n^d$, the total variation mixing time is of order $n^d \log n$, and the uniform mixing time is of order $n^{d+2}$. In particular, these three quantities are of different orders of magnitude.

Biased Random Walk on Lamplighter Groups and Graphs

Lyons, Pemantle, and Peres(4) found an example of a group on which a simple random walk has sublinear speed but an inward biased walk has linear speed. We examine a similar example by introducing a martingale that allows us to compute the exact speed of the inward biased walk. We apply the martingale techniques to the example of Lyons et al., and improve on their speed estimates for small amounts of bias. Finally, we examine the speed on a related family of graphs and construct an example on which the speed of the walk has a second phase change.