Infinite canonical super-Brownian motion and scaling limits

TL;DR

This paper introduces the infinite canonical super-Brownian motion (ICSBM), a measure-valued process serving as a universal scaling limit for critical, mean-field, infinite branching models in high dimensions, with proofs for specific models and conjectures for others.

Contribution

The paper constructs ICSBM and demonstrates it as the scaling limit for oriented percolation and branching random walk in high dimensions, proposing it as a universal limit for various models.

Findings

ICSBM is the scaling limit of high-dimensional oriented percolation incipient infinite cluster.

ICSBM is the scaling limit of incipient infinite branching random walk in any dimension.

Conjectures that ICSBM arises as the limit in other high-dimensional models like lattice trees and invasion percolation.

Abstract

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking super-Brownian to scaling limits in statistical mechanics.