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

TL;DR

The paper studies the connectedness of the supports of super-tree random measures (STRMs), including a detailed link to super-Brownian motion (SBM). It develops a general STRM framework with a discrete-time branching structure and spatial displacements, and then specializes to B-ary classical STRMs to obtain tractable geometric descriptions of the support. It proves that in high enough dimension the STRM support is almost surely totally disconnected, while percolation phenomena can create non-trivial connected components in certain parameter regimes (notably in 2D under a site-percolation bound). A key contribution is the precise connection between SBM and STRM; the results supply TD and percolation criteria that illuminate the corresponding SBM questions, contributing to the understanding of the geometric structure of SBM's support in dimensions 2 and 3. The work also leverages fractal percolation methods to bound percolation probabilities and to relate STRM behavior to well-studied random Cantor sets.

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 $ν$ 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 ($\mathrm{supp}(ν)$) for a particular choice of random displacement. Our main results are distinct sufficient conditions for the a.s. total disconnectedness (TD) of $\mathrm{supp}(ν)$, and for percolation on $\mathrm{supp}(ν)$ which will imply a.s. existence of a non-trivial connected component in $\mathrm{supp}(ν)$. 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.

Figures (5)

• Figure 1: Under the cluster law $\mathbb{P}^*$ the past trajectories of all the particles "present at time $1$ in $H_1$" can be described as those of a particle system that follow independent Brownian motions and split into two at time any $t$ with rate $\lambda(t)=\frac{1}{1-t}$. Note that at any time $t<1$, this system only contains a finite number of particles. Proposition \ref{['bpshbm']} ensures that we can recover the random measure $H_1$, historical Brownian motion at time $1$, from the trajectories of this particle system on $[0,1)$. We then take snapshots of the positions of the particles in this process at times $1-\mu^{-n}$ for $n\geq 1$. Theorem \ref{['thm:SBMSTRM']} ensures that the evolution of this process along this discrete sequence of times indeed fits into the STRM framework.
• Figure 2: For some site $x\in G_m$ on the grid, the cube $C_m(x)$ is the cube of edge-length $B^{-m}$ whose point with lowest coordinates is $x$.
• Figure 3: Black cubes correspond to positions $x$ on the grid with at least one particle whereas white cubes corresponds to positions with no particles. The union of the black cubes decreases to $\mathrm{supp} \ \nu$ as $m\rightarrow \infty$, see Lemma \ref{['Cantor']}.
• Figure 4: From left to right, examples of $\ell$-neighbour cubes, for $\ell = 2,1,0$. The intersection of their boundary is shown in red.
• Figure 5: On the event $E_m$, the special particle is alone at $S_m$ and at time $m+2$ all its descendants are at position $S_m+B^{-(m+1)} \vec{1}$. This ensures that all the cubes surrounding $C_{m+2}(S_{m+2})$ are going to remain empty.

Theorems & Definitions (68)

• Proposition 1.1
• Proposition 1.2
• Remark 1.3
• Theorem 1.4
• Definition
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9