Uniqueness for a class of one-dimensional stochastic PDEs using moment duality

1999/03/28 by Athreya, Siva, Tribe, Roger · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.math/9903194

Abstract

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^-3/2, plus an error term of order n-3/2-ε with ε> 0. This is a strong statement that the critical exponent δis given by δ= 2.

Cited by

• Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift

Related

• The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents
• The incipient infinite cluster in high-dimensional percolation
• Criticality, universality, and isoradiality
• Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
• Lattice trees, percolation and super-Brownian motion