Related papers: Conductance of a subdiffusive random weighted tree

Given a Galton-Watson process conditioned to have total progeny equal to $n$, we study the asymptotic probability that this conditioned Galton-Watson process has distance to the border bigger or equal than $k$, as the number of nodes $n…

This work proves new probability bounds relating to the height, width, and size of Galton-Watson trees. For example, if $T$ is any Galton-Watson tree, and $H$, $W$, and $|T|$ are the height, width, and size of $T$, respectively, then $H/W$…

We consider the simple random walk on Galton-Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return…

We study the random planar map obtained from a critical, finite variance, Galton-Watson plane tree by adding the horizontal connections between successive vertices at each level. This random graph is closely related to the well-known causal…

We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic…

We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals reproduce independently of the others and in…

We study random walk among random conductance (RWRC) on complete graphs with N vertices. The conductances are i.i.d. and the sum of conductances emanating from a single vertex asymptotically has an infinitely divisible distribution…

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical…

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, converges as $\eps\downarrow 0$ in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring…

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random;…

Let $\tau$n be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the n-th generation and (an, n $\in$ N *) is a deterministic positive sequence. We study…

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive…

We introduce the notion of a hereditary property for rooted real trees and we also consider reduction of trees by a given hereditary property. Leaf-length erasure, also called trimming, is included as a special case of hereditary reduction.…

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…

Aldous, Evans and Pitman (1998) studied the behavior of the fragmentation process derived from deleting the edges of a uniform random tree on $n$ labelled vertices. In particular, they showed that, after proper rescaling, the above…

Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton-Watson trees. For example, let $\mathcal{T}_1$ be the…

We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and…

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation…

We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus on the case of subtrees of the Continuum Random Tree generated by uniform sampling of…

For a complex number $\alpha$, we consider the sum of the $\alpha$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(\alpha)$ have been determined for $\Re\alpha…