Boolean percolation on digraphs and random exchange processes (Q6617593)

The paper investigates a long-range Boolean percolation model on directed graphs, which was introduced in [\textit{J. Lamperti}, J. Appl. Probab. 7, 89--98 (1970; Zbl 0196.18801)], and which is defined as follows: Let \(G=(V,E)\) be a directed infinite graph. Consider a collection of i.i.d.~random variables \((Y_x)_{x\in V}\) taking values in \(\mathbb N_0:=\{0,1,\dots\}\) distributed according to some measure \(\mu \) satisfying a non-triviality assumption \(\mu_0 \in (0,1) \). The occupied set \(V_\mu \) of this percolation model is the union of balls with radius given by \(Y_x\)'s,\N\[\NV_\mu = \bigcup_{x\in V}\{y\in V: d(x,y) < Y_x\},\N\]\Nwhere \(d\) is the directed distance on \(G\).\N\NThe properties of this percolation process are related to the so-called random exchange process which is a Markov chain \((X_n)_{n \ge 0}\) defined by \(X_0 = Y_0\) and\N\[\NX_{n+1} = \max \{X_n -1 , Y_{n+1}\}, \quad n\ge 0,\N\]\Nwhere \((Y_n)_{n\ge 0}\) are i.i.d.~with marginal \(\mu \). The connection between the two models is originally due to [\textit{M. P. W. Zerner}, Electron. J. Probab. 23, Paper No. 27, 24 p. (2018; Zbl 1390.60263)].\N\NThe main results of the paper can be briefly summarised as follows:\N\N1. When \(V = \mathbb N_0^n \) and \(E\) is the set of all pairs \((x,x+e_j)\) where \((e_j)_{j=1,\dots, n}\) is the usual basis of \(\mathbb R^n\), then the set of unoccupied vertices is a.s.~finite iff \((X_n)_{n\ge 0}\) is transient. The paper also gives a condition on the distribution \(\mu \) which is equivalent to transience:\N\[\N\sum_{m\ge 0} \prod_{k=1}^m \sum_{l=0}^{k-1}\mu_l < \infty.\N\]\N2. When \(V\) is a rooted \(n\)-nary tree with edges directed away from the root, then the occupied set \(V_\mu \) contains a path of infinite length iff the largest eigenvalue of a certain matrix (related to the random exchange process) is larger than \(n^{-1}\). In this case the occupied set contains an infinite number of infinite components. Otherwise, all components of the occupied set are finite. The proof of this result uses a connection to multi-type branching processes.\N\NFinally, the paper presents some examples.