The phase transition for parking on Galton-Watson trees

If, as above, the first arm to be activated belongs to the particle i and grabs particle j, then at time 1, the particle j (together with its arms) is shifted at the right-end of

For a diffusion in random potential, the recent [8] proves the Einstein relation by following the strategy of [11], adding to it a good control (uniform in the environment) of

(this procedure is defined when there is no Brownian part in the L´evy process with index given by the branching mechanism ψ). This random variable represents the time at which

We then apply this condition to get new results in the critical case (with a general offspring distribution) and in the sub-critical cases (with a generic offspring distribution) on

This result with Proposition 4.6 and Corollary 5.9 in [1] ends the proof of Theorem 1.2 for the case A = N and gives a complete description of the asymptotic distribution of

Our investigation treats the average number of transversals of fixed size, the size of a random transversal as well as the probability that a random subset of the vertex set of a

We say that a specific pattern M occurs in a tree T if M occurs in T as an induced subtree in the sense that the node degrees for the internal (filled) nodes in the pattern match

Notice that the previous statement immediately implies point (iii) in the statement of Theorem 1... One way of seeing this is to use Skorokhod’s representation theorem to exhibit