2 research outputs found
Local convergence of large critical multi-type Galton-Watson trees and applications to random maps
We show that large critical multi-type Galton-Watson trees, when conditioned
to be large, converge locally in distribution to an infinite tree which is
analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven
when we condition on the number of vertices of one fixed types, and with an
extra technical assumption if we count at least two types. We then apply these
results to study local limits of random planar maps, showing that large
critical Boltzmann-distributed random maps converge in distribution to an
infinite map.Comment: Corrected typoes, and a new proof of Lemma 4.
Random and cooperative sequential adsorption
Irreversible random sequential adsorption (RSA) on lattices, and continuum car parking analogues, have long received attention as models for reactions on polymer chains, chemisorption on single-crystal surfaces, adsorption in colloidal systems, and solid state transformations. Cooperative generalizations of these models (CSA) are sometimes more appropriate, and can exhibit richer kinetics and spatial structure, e.g., autocatalysis and clustering. The distribution of filled or transformed sites in RSA and CSA is not described by an equilibrium Gibbs measure. This is the case even for the saturation jammed state of models where the lattice or space cannot fill completely. However exact analysis is often possible in one dimension, and a variety of powerful analytic methods have been developed for higher dimensional models. Here we review the detailed understanding of asymptotic kinetics, spatial correlations, percolative structure, etc., which is emerging for these far-from-equilibrium processes