Related papers: Weighted-amenability and percolation
Weak amenability of a weighted group algebra, or a Beurling algebra, is a long-standing open problem. The commutative case has been extensively investigated and fully characterized. We study the non-commutative case. Given a weight function…
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of…
A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding…
Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24].…
In the past two decades, various properties of randomly perturbed/augmented (hyper)graphs have been intensively studied, since the model was introduced by Bohman, Frieze and Martin in 2003. The model usually considers a deterministic graph…
Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together…
We study the distribution of finite clusters in slightly supercritical ($p \downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if $G$ is a transitive nonamenable graph satisfying the…
Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite…
Inspired by biological evolution, we consider the following so-called accessibility percolation problem: The vertices of the unoriented $n$-dimensional binary hypercube are assigned independent $U(0, 1)$ weights, referred to as fitnesses. A…
We prove that for recurrent, reversible graphs, the following conditions are equivalent: (a) existence and uniqueness of the potential kernel, (b) existence and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack…
The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions…
We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field $K$. To capture the features of classical amenability that induce the vanishing of real bounded…
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 that \[\mathbf{P}_p(n \leq |K| <…
The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the…
Gr\"otschel, Lov\'asz, and Schrijver generalized the Lov\'asz $\vartheta$ function by allowing a weight for each vertex. We provide a similar generalization of Duan, Severini, and Winter's $\tilde{\vartheta}$ on non-commutative graphs.…
Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one…
Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(G), the probability that the cluster of the origin is finite but has a large…
We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters…
We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which…
For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic…