A Note on Truncated Long-Range Percolation with Heavy Tails on Oriented Graphs

Abstract

We consider oriented long-range percolation on a graph with vertex set Z d × Z + and directed edges of the form (x, t), (x + y, t + 1) , for x, y in Z d and t ∈ Z + . Any edge of this form is open with probability p y , independently for all edges. Under the assumption that the values p y do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on Z d .

Truncation of long-range percolation models with square non-summable interactions

“…, d}, n ∈ Z}. Proof : This proof is similar to the proof of Theorem 1 of Alves et al (2017). It consists in to define a family of special events, showing that they induce a supercritical oriented percolation process on an appropriate renormalized lattice, isomorphic to a subset of Z 2 + .…”

“…In each section, we will consider the truncation question on a different type of graph. In Section 3, we study the truncation question on a special oriented graph, generalizing the result of Theorem 1 of Alves et al (2017).…”

Truncation of long-range percolation model with square non-summable interactions

“…Proof. This proof is similar to the proof of Theorem 1 of [4]. It consists of two parts, the first one is to define a family of special events and the second part is to couple an exploration processes in an appropriate renormalized lattice, isomorphic to a subset of Z 2 + , using ideas of Grimmett and Marstrand [16].…”

“…for all d 2. The case d = 1 is an open question and a partial answer was given in [4], more precisely lim K→∞ P K {(0, 0) ∞} = 1 holds in d = 1 if lim sup n→∞ p n > 0. The next theorem improve the result of Theorem 1 of [4] replacing the hypothesis lim sup n→∞ p n > 0 by n p 2 n = ∞ (that is, some sequences (p n ) n decaying to zero are allowed like…”

“…In each section, we will consider the truncation question in a different type of graph. In Section 2, we study the truncation question on a special oriented graph, generalizing the result of Theorem 1 of [4].…”