Fabio Marcello Lopes: Percolation for Poisson Bipartite stable graphs in R^d

Let R and M be two independent homogeneous Poisson processes withfinite intensities λ_R and λ_M, respectively.

Assign independently to each point in R and M a random number of half-edges with laws X and Y, respectively. Both are positive integer-valued random variables. We construct translation-invariant simple graphs in ℝ^d in which a.s. all half-edges are paired and we do not allow pairings between points of the same colour. We call such graphs of two-colour multi-matchings.

We will introduce this model and a translation-invariant scheme to pair the half-edges that generates a stable graph in the sense of the Gale-Shapley stable marriage, this scheme generalizes the two-colour stable matching of Holroyd, Pemantle, Peres and Schramm 09'. Also, we discuss some percolative properties of this model and compare them with those of its one-colour version studied in Deijfen, Häggström and Holroyd 11' and Deijfen, Holroyd and Peres 11'.