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Infinite Canonical Super-Brownian Motion and Scaling Limits

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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 \(\mathbb{Z}^d\) when these objects are critical, mean-field and 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 above 6 dimensions, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.

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  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600, MB, Eindhoven, The Netherlands

    Remco van der Hofstad

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  1. Remco van der Hofstad
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Communicated by M. Aizenman

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van der Hofstad, R. Infinite Canonical Super-Brownian Motion and Scaling Limits. Commun. Math. Phys. 265, 547–583 (2006). https://doi.org/10.1007/s00220-006-0044-y

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