Related papers: The phase transition for parking on Galton--Watson…

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on its outdegree. We give a complete picture of the local convergence of critical or sub-critical marked Galton-Watson…

We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on itsout-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level n -- 1…

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the…

We study $I(T)$, the number of inversions in a tree $T$ with its vertices labeled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of $I(T)$ have explicit formulas involving the…

We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is…

We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution has finite covariance matrices. Our study relies on an ancestral…

We study $S(\mathcal T_{n})$, the number of subtrees in a conditioned Galton--Watson tree of size $n$. With two very different methods, we show that $\log(S(\mathcal T_{n}))$ has a Central Limit Law and that the moments of $S(\mathcal…

Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with types coming from a finite alphabet, conditioned to non-extinction. The lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk which, when at a…

The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in $\mathbb{R}^d$ with the formal Hamiltonian $H(\omega)=\text{Volume}(\cup_{x\in\omega} B_1(x))$, where $\omega$ is a locally finite configuration of points…

The aim of this lecture is to give an overview of old and new resultson Bienaym\'e-Galton-Watson (BGW) trees. After introducing the framework of discretetrees, we first give alternative proofs of classical results on theextinction…

A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is…

We study phase-transitions of the ferromagnetic $q$-state Potts chain with random nearest-neighbour couplings having a variance $\Delta^2$ and with homogeneous long-range interactions, which decay with the distance as a power…

We work on a Galton--Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$-th level of the tree, as $n$ goes to infinity, by a mostly analytic…

A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure…

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 consider the number and distribution of minima in random landscapes defined on non-Euclidean lattices. Using an ensemble where random landscapes are reweighted by a fugacity factor $z$ for each minimum they contain, we construct first a…

We study the semi-infinite Ising model with an external field $h_i = \lambda |i_d|^{-\delta}$, $\lambda$ is the wall influence, and $\delta>0$. This external field decays as it gets further away from the wall. We are able to show that when…

We use Monte Carlo techniques and analytical methods to study the phase diagram of the M--component Widom-Rowlinson model on the bcc-lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion…

We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite…

We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit…