Unlacing the lace expansion: a survey to hypercube percolation

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers [9,10,11,24]. Secondly, we explain how this study can be performed without the use of the so-called "lace-expansion" technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. We hope that some of these techniques will be useful to obtain non-perturbative proofs in the analogous, yet much more difficult study on high-dimensional tori.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1201.395

Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in \Zd, oriented percolation in \Zd\times\Zp, and the contact process in \Zd, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point \pc. We investigate the value of \pc when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that \pc=1+C(D)+O(L^{-2d}), where the universal term 1 is the mean-field critical value, and the model-dependent term $C(D)=O(L^{-d})$ is written explicitly in terms of the function $D$. Our proof is based on the lace expansion for each of these models.Comment: 22 pages, no figure

Expansion in high dimension for the growth constants of lattice trees and lattice animals

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the expansion, making the present paper more self-contained with very little change to the structure of the original paper. Accepted for publication in Combinatorics Probability and Computin

Expansion for the critical point of site percolation: the first three terms

We expand the critical point for site percolation on the $d$-dimensional hypercubic lattice in terms of inverse powers of $2d$, and we obtain the first three terms rigorously. This is achieved using the lace expansion.Comment: 22 page

Expansion of percolation critical points for Hamming graphs

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let be the degree and be the number of vertices of H(d, n). Let be the critical point for bond percolation on H(d, n). We show that, for fixed and , which extends the asymptotics found in [10] by one order. The term is the width of the critical window. For we have , and so the above formula represents the full asymptotic expansion of . In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for . The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random grap

A branching process with deletions and mergers that matches the threshold for hypercube percolation

We define a graph process G(p,q) based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube Qd and the lattice Zd for large d. Individuals have Poisson offspring distribution with mean 1+p and certain deletions and mergers occur with probability q; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively.We prove survival and extinction under certain conditions on p and q that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice Zd and the hypercube Qd. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold. However, it can provide a conjecture for other high-dimensional, odd-cycle free transitive graphs such as the body-centered cubic lattice.The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of G(p,q) is monotone in p or q; we discuss this and some other open problems regarding the new graph process

A branching process with deletions and mergers that matches the threshold for hypercube percolation

We define a graph process G(p,q) based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube Qd and the lattice Zd for large d. Individuals have Poisson offspring distribution with mean 1+p and certain deletions and mergers occur with probability q; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively.We prove survival and extinction under certain conditions on p and q that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice Zd and the hypercube Qd. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold. However, it can provide a conjecture for other high-dimensional, odd-cycle free transitive graphs such as the body-centered cubic lattice.The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of G(p,q) is monotone in p or q; we discuss this and some other open problems regarding the new graph process