Nonamenable Graphs Research Articles

Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters

Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters

Factor-of-iid balanced orientation of non-amenable graphs

We show that if a non-amenable, quasi-transitive, unimodular graph G has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs.As a consequence, we also obtain that when G is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.

Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi- $$\kappa $$ -to-one in a natural sense for some $$\kappa >0$$ . For non-amenable graphs, all quasi-isometries are quasi- $$\kappa $$ -to-one for any $$\kappa >0$$ , while for amenable ones there exists at most one possible such $$\kappa $$ . For an amenable graph X, we show that the set of possible $$\kappa $$ forms a subgroup of $$\mathbb {R}_{>0}$$ that we call the (measure-)scaling group of X. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a given locally compact group have same scaling groups. We compute the scaling group in a number of cases. For instance it is all of $$\mathbb {R}_{>0}$$ for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup $$\mathbb {Q}_{>0}$$ for lamplighter groups over finitely presented amenable groups.

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p_c(G) then there exists a positive constant c_p such thatPp(n≤|K|<∞)≤e-cpn\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf {P}_p(n \\le |K| < \\infty ) \\le e^{-c_p n} \\end{aligned}$$\\end{document}for every nge 1, where K is the cluster of the origin. We deduce the following two corollaries:Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999).For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

The $L^{2}$ boundedness condition in nonamenable percolation

Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities $T_{p_{c}}(u,v)=\mathbb {P}_{p_{c}}(u\leftrightarrow v)$ is bounded as an operator $T_{p_{c}}:L^{2}(V)\to L^{2}(V)$ and proved that this conjecture holds for several classes of graphs, including all transitive, nonamenable, Gromov hyperbolic graphs. In notation, the conjecture states that $p_{c}<p_{2\to 2}$, where for each $q\in [1,\infty ]$ we define $p_{q\to q}$ to be the supremal value of $p$ for which the operator norm $\|T_{p}\|_{q\to q}$ is finite. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour. In this paper we further investigate the consequences of the $L^{2}$ boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all $p<p_{2\to 2}$; ii) If $p_{c}<p_{2\to 2}$, then the critical exponent governing the extrinsic diameter of a critical cluster is $1$; iii) Below $p_{2\to 2}$, percolation is “ballistic" in the sense that the intrinsic (a.k.a. chemical) distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If $p_{c}<p_{2\to 2}$, then $\|T_{p_{c}}\|_{q\to q} \asymp (q-1)^{-1}$ and $p_{q\to q}-p_{c} \asymp q-1$ as $q\downarrow 1$; v) If $p_{c}<p_{2\to 2}$, then various ‘multiple-arm’ events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order $(p-p_{c})^{3}$ as $p \downarrow p_{c}$. All of these results are new even in the Gromov hyperbolic case. Finally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.

Active phase for activated random walks on $\mathbb{Z}^{d}$, $d\geq3$, with density less than one and arbitrary sleeping rate

Il a ete conjecture que la densite critique pour le modele de marches aleatoires activees etait strictement inferieur a 1 pour toute valeur du taux d’endormissement. Nous demontrons cette conjecture pour $\mathbb{Z}^{d}$ quand $d\geq3$ et, plus generalement, pour les graphes sur lesquels la marche aleatoire est transitoire. De plus, nous montrons l’existence d’une transition de phase pour les graphes non moyennables, generalisant ainsi des resultats anterieurs qui demandaient que le graphe soit moyennable ou un arbre regulier.

Critical Phase in Complex Networks: a Numerical Study

We compare phase transition and critical phenomena of bond percolation on Euclidean lattices, nonamenable graphs, and complex networks. On a Euclidean lattice, percolation shows a phase transition between the nonpercolating phase and percolating phase at the critical point. The critical point is stretched to a finite region, called the critical phase, on nonamenable graphs. To investigate the critical phase, we introduce a fractal exponent, which characterizes a subextensive order of the system. We perform the Monte Carlo simulations for percolation on two nonamenable graphs - the binary tree and the enhanced binary tree. The former shows the nonpercolating phase and the critical phase, whereas the latter shows all three phases. We also examine the possibility of critical phase in complex networks. Our conjecture is that networks with a growth mechanism have only the critical phase and the percolating phase. We study percolation on a stochastically growing network with and without a preferential attachment mechanism, and a deterministically growing network, called the decorated flower, to show that the critical phase appears in those models. We provide a finite-size scaling by using the fractal exponent, which would be a powerful method for numerical analysis of the phase transition involving the critical phase.

Localization for linearly edge reinforced random walks

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.

Bernoulli and self-destructive percolation on non-amenable graphs

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as $p\searrow p_c$. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.

HARMONIC ANALYSIS ON CAYLEY TREES II: THE BOSE–EINSTEIN CONDENSATION

We investigate the Bose–Einstein Condensation on non-homogeneous non-amenable networks for the model describing arrays of Josephson junctions. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. The resulting topological model, whose Hamiltonian is the pure hopping one given by the opposite of the adjacency operator, has also a mathematical interest in itself. The present paper is then the application to the Bose–Einstein Condensation phenomena, of the harmonic analysis aspects, previously investigated in a separate work, for such non-amenable graphs. Concerning the appearance of the Bose–Einstein Condensation, the results are surprisingly in accordance with the previous ones, despite the lack of amenability. The appearance of the hidden spectrum for low energies always implies that the critical density is finite for all the models under consideration. We also show that, even when the critical density is finite, if the adjacency operator of the graph is recurrent, it is impossible to exhibit temperature states which are locally normal (i.e. states for which the local particle density is finite) describing the condensation at all. A similar situation seems to occur in the transient cases for which it is impossible to exhibit locally normal states ω describing the Bose–Einstein Condensation with mean particle density ρ(ω) strictly greater than the critical density ρc. Indeed, it is shown that the transient cases admit locally normal states exhibiting Bose–Einstein Condensation phenomena. In order to construct such locally normal temperature states by infinite volume limits of finite volume Gibbs states, a careful choice of the sequence of the chemical potentials should be done. For all such states, the condensate is essentially allocated on the base point supporting the perturbation. This leads to ρ(ω) = ρc as the perturbation is negligible with respect to the whole network. We prove that all such temperature states are Kubo–Martin–Schwinger states for the natural dynamics associated to the (formal) pure hopping Hamiltonian. The construction of such a dynamics, which is a delicate issue, is also provided in detail.

Percolation on Infinite Graphs and Isoperimetric Inequalities

We consider the Bernoulli bond percolation process (with parameter $p$) on infinite graphs and we give a general criterion for bounded degree graphs to exhibit a non-trivial percolation threshold based either on a single isoperimetric inequality if the graph has a bi-infinite geodesic, or two isoperimetric inequalities if the graph has not a bi-infinite geodesic. This new criterion extends previous criteria and brings together a large class of amenable graphs (such as regular lattices) and non-amenable graphs (such trees). We also study the finite connectivity in graphs satisfying the new general criterion and show that graphs in this class with a bi-infinite geodesic always have finite connectivity functions with exponential decay as $p$ is sufficiently close to one. On the other hand, we show that there are graphs in the same class with no bi-infinite geodesic for which the finite connectivity decays sub-exponentially (down to polynomially) in the highly supercritical phase even for $p$ arbitrarily close to one.

Non-amenable Cayley graphs of high girth have $p_c &lt; p_u$ and mean-field exponents

In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.

Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees

We perform Monte Carlo simulations to study the Bernoulli (p) bond percolation on the enhanced binary tree which belongs to the class of nonamenable graphs with one end. Our numerical results show that the system has two distinct percolation thresholds pc1 and pc2. The mean cluster size diverges as p approaches pc1 from below. The system is critical at all the points in the intermediate phase (pc1 < p < pc2) and there exist infinitely many infinite clusters. In this phase, the corresponding fractal exponent continuously increases with p from zero to unity. Above pc2 the system has a unique infinite cluster.

Interlacement percolation on transient weighted graphs

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value $u_*$ for the percolation of the vacant set is finite. We also prove that, once $\mathcal{G}$ satisfies the isoperimetric inequality $I S_6$ (see (1.5)), $u_*$ is positive for the product $\mathcal{G} \times \mathbb{Z}$ (where we endow $\mathbb{Z}$ with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value $u_*$.

Internal Diffusion-Limited Aggregation on non-amenable graphs

The stochastic growth model Internal Diffusion Limited Aggregation was defined in 1991 by Diaconis and Fulton. Several shape results are known when the underlying state space is the d-dimensional lattice, or a discrete group with exponential growth. We prove an extension of the shape result of Blachere and Brofferio for Internal Diffusion Limited Aggregation on a wide class of Markov chains on non-amenable graphs.

Glauber dynamics on nonamenable graphs: boundary conditions and mixing time

We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growing with the inverse temperature. In addition, we construct a graph in this class, for which the mixing time in the free boundary case is exponentially large in n. This provides a first example where the mixing time jumps from exponential to linear in n while passing from free to (+)-boundary condition. These results extend the analysis of Martinelli, Sinclair and Weitz to a wider class of nonamenable graphs.

A phase transition for the metric distortion of percolation on the hypercube

Let H n be the hypercube {0, 1} n , and denote by H n,p Bernoulli bond percolation on H n , with parameter p = n −α . It is shown that at α = 1/2 there is a phase transition for the metric distortion between H n and H n,p . For α < 1/2, the giant component of H n,p is likely to be quasi-isometric to H n with constant distortion (depending only on α). For 1/2 < α < 1 the minimal distortion tends to infinity as a power of n. We argue that the phase 1/2 < α < 1 is an analogue of the non-uniqueness phase appearing in percolation on non-amenable graphs.

Percolation on nonunimodular transitive graphs

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with pc=1 on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.

Mean-Field Criticality for Percolation on Planar Non-Amenable Graphs

The critical exponents β, γ, δ and Δ are proved to exist and to take their mean-field values for independent percolation on the following classes of infinite, locally finite, connected transitive graphs: (1) Non-amenable planar with one end. (2) Unimodular with infinitely many ends.

Explicit Isoperimetric Constants and Phase Transitions in the Random-Cluster Model

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q \geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value and examples of planar regular graphs with regular dual where $p_ \mathrm{c}^{\mathrm{free}} (q) > p_ \mathrm{u}^{\mathrm{wired}} (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove nonrobust phase transition for the Potts model on nonamenable regular graphs.