Sizes of the largest clusters for supercritical percolation on random recursive trees

Sizes of the largest clusters for supercritical percolation on random recursive trees

We consider Bernoulli bond-percolation on a random recursive tree of size n ≫ 1, with supercritical parameter p(n) = 1− t/ ln n+ o(1/ ln n) for some t > 0 fixed. We show that with high probability, the largest cluster has size close to e−tn whereas the next largest clusters have size of order n/ lnn only and are distributed according to some Poisson random measure.

Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium me...

Scaling limit of simple random walk on supercritical percolation clusters

Branches in random recursive k-ary trees

In this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. We also find the expectation of the number of nodes of a given outdegree in a branch of such trees.

On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees

We consider a Bernoulli bond percolation on a random recursive tree of size n ≫ 1, with supercritical parameter pn = 1 − c/ lnn for some c > 0 fixed. It is known that with high probability, there exists then a unique giant cluster of size Gn ∼ e−c, and it follows from a recent result of Schweinsberg [22] that Gn has non-gaussian fluctuations. We provide an explanation of this by analyzing the e...