A Boltzmann Approach to Percolation on Random Triangulations

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length$n$decays exponentially with$n$except at a particular value$p_{c}$of the percolation parameter$p$for which the decay is polynomial (of order$n^{-10/3}$). Moreover, the probability that the origin cluster has size$n$decays exponentially if$p<p_{c}$and polynomially if$p\geqslant p_{c}$.The critical percolation value is$p_{c}=1/2$for site percolation, and$p_{c}=(2\sqrt{3}-1)/11$for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at$p_{c}$, the percolation clusters conditioned to have size$n$should converge toward the stable map of parameter$\frac{7}{6}$introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = pn ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter pn tending to 0.

The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview

International audience Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.

FINITE-SIZE SCALING IN TWO-DIMENSIONAL CONTINUUM PERCOLATION MODELS

We test the universal finite-size scaling of the cluster mass order parameter in two-dimensional (2D) isotropic and directed continuum percolation models below the percolation threshold by computer simulations. We found that the simulation data in the 2D continuum models obey the same scaling expression of mass M to sample size L as generally accepted for isotropic lattice problems, but with a positive sign of the slope in the ln–ln plot of M versus L. Another interesting aspect of the finite-size 2D models is also suggested by plotting the normalized mass in 2D continuum and lattice bond percolation models versus an effective percolation parameter, independent of the system structure (i.e., lattice or continuum) and of the possible directions allowed for percolation (i.e., isotropic or directed) in regions close to the percolation thresholds. Our study is the first attempt to map the scaling behavior of the mass for both lattice and continuum model systems into one curve.

GRADIENT PERCOLATION AND FRACTAL FRONTIERS IN IMAGE PROCESSING

In contrast to standard percolation where criticality is reached only for a particular value pc of the driving parameter p, gradient percolation exists without the precise tuning of a percolation parameter. For this reason it may be a common physical situation. Very generally, gradient percolation will appear in a uniform system whenever there exists a local random response to an excitation which varies in space. We show that such a situation exists in the example of photographic imaging, due to the random aspect of the photographic process. In this case gradient percolation may be used as a filter for recovering fuzzy images. This filter has the advantage of self-adjusting and to be neutral in regard to the size of the objects. In particular it could be used to increase artificially the depth of focus on photographs that are partially fuzzy.