Related papers: A homological upper bound on critical probabilitie…
Recent research on percolation has led to the construction of an infinite class of lattices for which the percolation thresholds can be determined exactly. We discuss the mathematical basis for the solutions of bond percolation models, and,…
An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented.
We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence…
We consider Bernoulli bond percolation on the product graph of a regular tree and a line. Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $\alpha(p)$. The function $\alpha(p)$ is…
We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The…
Correlations are known to play a crucial role in determining the structure of complex networks. Here we study how their presence affects the computation of the percolation threshold in random hypergraphs. In order to mimic the correlation…
We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper…
We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops…
We give an upper bound for the uniqueness transition on an arbitrary locally finite graph ${\cal G}$ in terms of the limit of the spectral radii $\rho\left[ H({\cal G}_t)\right]$ of the non-backtracking (Hashimoto) matrices for an…
Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and…
We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $\Lambda$ by $\ell$ bonds connecting the same adjacent vertices,…
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| <…
Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1…
In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain…
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…
Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging…
Majority bootstrap percolation is a model of infection spreading in networks. Starting with a set of initially infected vertices, new vertices become infected once half of their neighbours are infected. Balogh, Bollob\'{a}s and Morris…
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized…
We prove infinitely many cases of conjectured sharp upper and lower bounds for the spanning tree entropy of any planar lattice graph. These bounds come from volumes of associated hyperbolic alternating links, right-angled hyperbolic…
We study bond percolation on the hypercube $\{0,1\}^m$ in the slightly subcritical regime where $p = p_c (1-\varepsilon_m)$ and $\varepsilon_m = o(1)$ but $\varepsilon_m \gg 2^{-m/3}$ and study the clusters of largest volume and diameter.…