Mean-field lattice trees

Abstract

We introduce a mean-field model of lattice trees based on embeddings into $\Z^d$ of abstract trees having a critical Poisson offspring distribution. This model provides a combinatorial interpretation for the self-consistent mean-field model introduced previously by Derbez and Slade, and provides an alternate approach to work of Aldous. The scaling limit of the mean-field model is integrated super-Brownian excursion (ISE), in all dimensions. We also introduce a model of weakly self-avoiding lattice trees, in which an embedded tree receives a penalty $e^{-\beta}$ for each self-intersection. The weakly self-avoiding lattice trees provide a natural interpolation between the mean-field model ($\beta=0$), and the usual model of strictly self-avoiding lattice trees ($\beta=\infty$) which associates the uniform measure to the set of lattice trees of the same size.