Mean-field lattice trees

1999/04/26 by Christian Borgs, Borgs, Christian, Jennifer Chayes +5
Computer Science · #Advanced Algebra and Logic #Advanced Database Systems and Queries #Data Management and Algorithms #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.math/9904184

openalex publication_date 1999/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^-β for each self-intersection. The weakly self-avoiding lattice trees provide a natural interpolation between the mean-field model (β=0), and the usual model of strictly self-avoiding lattice trees (β=∞) which associates the uniform measure to the set of lattice trees of the same size.

Related

• Correlated Drainage Model
• Entropy of self-avoiding branching polymers: mean field theory and Monte Carlo simulations
• Rare subtree patterns in size-conditioned Bienaymé trees: Poisson approximation and declumping
• Lattice trees, percolation and super-Brownian motion
• Scaling limit of critical random trees in random environment