Abstract A z -matching on a bipartite graph is set of edges, among which each vertex two types the adjacent to at most 1 and ( ⩾ 1 ) respectively. The problem concerns finding -matchings with maximum size. Our approach this com...

Given two graphs G and H, it is said that percolates in H-bootstrap process if one could join all the nonadjacent pairs of vertices some order such a new copy H created at each step. Balogh, Bollobás Morris 2012 investigated threshold percolation Erdős–Rényi model for complete graph proposed similar problem $$H=K_{s,t}$$ , bipartite graph. In this paper, we provide lower upper bounds on $$K_{2,...

We obtain the exact solution of the bond-percolation thresholds with inhomogeneous probabilities on the square lattice. Our method is based on the duality analysis with real-space renormalization, which is a profound technique invented in the spin-glass theory. Our formulation is a more straightforward way than that of a very recent study on the same problem [R. M. Ziff et al., J. Phys. A: Math...

A scaling analysis introduced by Halperin, Feng and Sen to estimate critical exponents for the electrical conductivity and elastic constant is extended to determine the critical behaviour of electrical and mechanical failure near the percolation threshold for a class of disordered continuum systems (Swiss-cheese models). Above the dimension dc = 3/2 for the electrical problem and at any dimensi...

The establishment and spreading of biological populations depends crucially on population growth at low densities. The Allee effect is a problem in those populations where the per capita growth rate at low densities is reduced. We examine stochastic spatial models in which the reproduction rate changes across a gradient g so that the population undergoes a 2D-percolation transition. Without the...

This paper presents some weighted H-regularity estimates for a model Poisson problem with discontinuous coefficient at high contrast. The coefficient represents a random particle reinforced composite material, i.e., perfectly conducting circular particles are randomly distributed in some background material with low conductivity. Based on these regularity results we study the percolation of the...

In this work we consider two first-passage percolation problems. In the first part we concern ourselves with effectively one-dimensional graphs with vertex set {1 . . . , n}×{0, 1} and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate χ by solving certain recursive distributional equations and invoking result...

We show that symmetry, represented by a graph’s automorphism group, can be used to greatly reduce the computational work for the substitution method. This allows application of the substitution method over larger regions of the problem lattices, resulting in tighter bounds on the percolation threshold pc. We demonstrate the symmetry reduction technique using bond percolation on the (3, 12) latt...

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large β). The main results are bounds on percolation thresholds ρ±(β) in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster...

We examine the tight-binding density of states of a random binary alloy in a square lattice. In the split-band limit, which is similar to the model used for studing the quantum percolation problem, numerical calculations show that there is a pseudogap at the center of the lower subband, near and beyond the geometrical percolation threshold. This behavior is studied in two ways, by analyzing the...