Total disconnectedness and percolation for the supports of super-tree random measures

TL;DR

This paper investigates the connectedness and disconnectedness of the support of super-tree random measures, providing conditions for total disconnectedness and percolation, with implications for super-Brownian motion in various dimensions.

Contribution

It establishes new sufficient conditions for total disconnectedness and percolation of STRM supports, linking these properties to super-Brownian motion and percolation theory.

Findings

Support is totally disconnected in dimensions three and higher.

Existence of a non-trivial connected component in two dimensions under certain conditions.

Connections between STRMs and super-Brownian motion suggest similar properties in related models.

Abstract

Super-tree random measures (STRMs) were introduced by Allouba, Durrett, Hawkes and Perkins as a simple stochastic model which emulates a superprocess at a fixed time. A STRM $\nu$ arises as the a.s. limit of a sequence of empirical measures for a discrete time particle system which undergoes independent supercritical branching and independent random displacement (spatial motion) of children from their parents. We study the connectedness properties of the closed support of a STRM ($\mathrm{supp}(\nu)$) for a particular choice of random displacement. Our main results are distinct sufficient conditions for the a.s. total disconnectedness (TD) of $\mathrm{supp}(\nu)$, and for percolation on $\mathrm{supp}(\nu)$ which will imply a.s. existence of a non-trivial connected component in $\mathrm{supp}(\nu)$. We illustrate a close connection between a subclass of these STRM's and super-Brownian motion (SBM). For these particular STRM's the above results give a.s. TD of the support in three and higher dimensions and the existence of a non-trivial connected component in two dimensions, with the three-dimensional case being critical. The latter two-dimensional result assumes that $p_c(\mathbb{Z}^2)$, the critical probability for site percolation on $\mathbb{Z}^2$, is less than $1-e^{-1}$. (There is strong numerical evidence supporting this condition although the known rigorous bounds fall just short.) This gives evidence that the same connectedness properties should hold for SBM. The latter remains an interesting open problem in dimensions $2$ and $3$ ever since it was first posed by Don Dawson over $30$ years ago.