Invasion percolation on the Poisson-weighted infinite tree

Abstract

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $\sigma\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary" representations of the Poisson incipient infinite cluster as random graphs on $\mathbb {Z}$ which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane $\mathbb {R}\times[0,\infty)$.Comment: Published in at http://dx.doi.org/10.1214/11-AAP761 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Short Paths for First Passage Percolation on the Complete Graph

“…In the setting of Theorem 1.1, F Y (y) = 1 − e −y 1/sn ≈ y 1/sn when y = y n tends to zero fast enough, so that u n (x) can be taken as u n (x) ≈ (x/n) sn (here ≈ indicates approximation with uncontrolled error). Then we see in (1.2) that W n − 1 λn log (n/s 3 n ) ≈ u n (M (1) ∨ M (2) ) for λ n = n sn Γ(1 + 1/s n ) sn , which means that the weight of the smallest-weight path has a deterministic part 1 λn log (n/s 3 n ), while its random fluctuations are of the same order of magnitude as some of the typical values for the minimal edge weight adjacent to vertices 1 and 2. For j ∈ {1, 2}, one can think of M (j) as the time needed to escape from vertex j.…”

“…H n − φ n log (n/s 3 n ) s 2 n log (n/s 3 n ) d −→ Z, (1.9) where Z is standard normal, and M (1) , M (2) are i.i.d. random variables for which P(M (j) ≤ x) is the survival probability of a Poisson Galton-Watson branching process with mean x.…”

“…The vertices of T (1) are given by finite sequences of natural numbers headed by the symbol ∅ 1 , which we write as ∅ 1 j 1 j 2 · · · j k . The sequence ∅ 1 denotes the root vertex of T (1) . We concatenate sequences v = ∅ 1 i 1 · · · i k and w = ∅ 1 j 1 · · · j m to form the sequence vw = ∅ 1 i 1 · · · i k j 1 · · · j m of length |vw| = |v| + |w| = k + m. Identifying a natural number j with the corresponding sequence of length 1, the j th child of a vertex v is vj, and we say that v is the parent of vj.…”

“…The Poisson-weighted infinite tree is an infinite edge-weighted tree in which every vertex has infinitely many (ordered) children. To describe it formally, we associate weights to the edges of T (1) . By construction, we can index these edge weights by non-root vertices, writing the weights as X = (X v ) v =∅ 1 , where the weight X v is associated to the edge between v and its parent p(v).…”

“…The following theorem, taken from [Part I, Theorem 2.6] and proved in [Part I, Section 4.3], establishes a close connection between FPP on K n and FPP on the PWIT with edge weights (f n (X v )) v∈τ : Theorem 2.6 (Coupling to FPP on PWIT). The law of (SWT (1) t ) t≥0 is the same as the law of π M BP (1) t t≥0…”