Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. In the talk, we will look at interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We will not only be interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. In particular, we want to know how sensitive the Poisson construction is to small changes in the initial state. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay, which are a sort of Markov chains in which time has a tree-like structure and in which the state of each vertex is a random function of its descendants. The abstract theory will be demonstrated on an example of a particle system with cooperative branching and deaths.
This is joint work with Anja Sturm and Tibor Mach.

Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that:

1. Every infinite cluster has anchored expansion (a relaxation of having positive Cheeger constant), and so is nonamenable in some weak sense. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
2. Various observables, including the percolation probability and the truncated susceptibility (which was not even known to be finite!) are analytic functions of p throughout the entire supercritical phase.
3. A RW on an infinite cluster returns to the origin at time 2n with probability exp(-Θ(n^1/3)).

Joint work with Tom Hutchcroft.

Tekintsünk egy elágazó bolyongást Z^d x Z₊-ban, ahol az utódeloszlás kritikus, azon feltétel mellett, hogy nem hal ki. Az elágazó bolyongás nyoma a meglátogatott (x,t) rácspontok által alkotott véletlen gráf. Jelölje R(n) az (o,0) origó es a t=n szint közötti elektromos ellenállást e gráfban. Ha d>6, akkor E[R(n)]≥cn, viszont ha d≤5, akkor E[R(n)] = O(n^1-a), ahol a > 0 [Jarai&Nachmias]. Itt a d=6 kritikus dimenziót vizsgáljuk, és megmutatjuk, hogy ekkor E[R(n)] = O( n log^-an).
(D. Mata Lopez-zel közös eredmény).