ABSTRACT:In (Aldous, Math. Proc. Cambridge Philos. Soc. 128 (2000), 465-477), Aldous constructed a growth process for the binary tree where clusters freeze as soon as they become infinite. It was pointed out by Benjamini and Schramm that such a process does not exist for the square lattice.This motivated us to investigate the modified process on the square lattice, where clusters freeze as soon as they have diameter larger than or equal to N, the parameter of the model. The non-existence result, mentioned above, raises the question if the N−parameter model shows some 'anomalous' behaviour as N → ∞. For instance, if one looks at the cluster of a given vertex, does, as N → ∞, the probability that it eventually freezes go to 1? Does this probability go to 0? More generally, what can be said about the size of a final cluster? We give a partial answer to some of such questions.

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.

In the Constrained-degree percolation model on a graph (V, E) there are a sequence, (U e ) e∈E , of i.i.d. random variables with distribution U [0, 1] and a positive integer k. Each bond e tries to open at time U e , it succeeds if both its end-vertices would have degrees at most k − 1. We prove a phase transition theorem for this model on the square lattice L 2 , as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.

In this note we consider long range $q$-states Potts models on $\mathbf{Z}^d$, $d\geq 2$. For various families of non-summable ferromagnetic pair potentials $\phi(x)\geq 0$, we show that there exists, for all inverse temperature $\beta>0$, an integer $N$ such that the truncated model, in which all interactions between spins at distance larger than $N$ are suppressed, has at least $q$ distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type $\phi(x)\sim \|x\|^{-\alpha}$, $0\leq\alpha\leq d$. These results are obtained using simple percolation arguments.Comment: 18 pages, 4 figure

We provide a lower bound for the convergence radius of the Mayer series of the LennardJones gas which strongly improves on the classical bound obtained by Penrose and Ruelle 1963. To obtain this result we use an alternative estimate recently proposed by Morais et al. (J. Stat. Phys. 2014) for a restricted class of stable and tempered pair potentials (namely those which can be written as the sum of a non-negative potential plus an absolutely integrable and stable potential) combined with a method developed by Locatelli and Schoen (J. Glob. Optim. 2002) for establishing a lower bound for the minimal interatomic distance between particles interacting via a Morse potential in a cluster of minimum-energy configurations.