On the size of the largest cluster in 2D critical percolation

We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is between an^2 pi(n) and bn^2 pi(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the argument.Comment: 12 pages, 3 figures, minor change

Random graph asymptotics on high-dimensional tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times $V^{2/3}$ and below by a small constant times $V^{2/3}(log V)^{-4/3}$, where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Z^d under which the lower bound can be improved to small constant times $V^{2/3}$, i.e., we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results by Borgs, Chayes, van der Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Z^d. We also strongly rely on mean-field results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade (1990) and Hara, van der Hofstad and Slade (2003).Comment: 22 page

Maximal clusters in non-critical percolation and related models

We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations are related to maxima of independent random variable with law described by a single cluster.Comment: 26 pages, no figure

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. The edge probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W, in which case the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power-law case, in which P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and some \tau>4 and c>0, the largest critical connected component in a graph of size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When, instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4) and c>0, the largest critical connected component is of the much smaller order n^{(\tau-2)/(\tau-1)}.Comment: 26 page

Upper bounds on the percolation correlation length

We study the size of the near-critical window for Bernoulli percolation on $\mathbb Z^d$. More precisely, we use a quantitative Grimmett-Marstrand theorem to prove that the correlation length, both below and above criticality, is bounded from above by $\exp(C/|p-p_c|^2)$. Improving on this bound would be a further step towards the conjecture that there is no infinite cluster at criticality on $\mathbb Z^d$ for every $d\ge2$.Comment: 21 pages, 2 figure

The giant in random graphs is almost local

Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many random graph properties do not directly converge when the random graph converges locally. A notable, and important, random graph property that does not follow from local convergence is the size and uniqueness of the giant component. We provide a simple criterion that guarantees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant are also described by the local limit. We apply this novel method to the configuration model as a proof of concept, reproving a result that is well-established. As a side result this proof, we show that the proof also implies the small-world nature of the configuration model.Comment: 23 page