Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$

TL;DR

This paper investigates long-range percolation in high dimensions, demonstrating that infinite clusters persist after truncating long edges under certain degree conditions, with implications for related Potts models.

Contribution

It establishes that in high-dimensional long-range percolation with infinite expected degree, infinite clusters survive after truncating edges beyond a certain length, extending understanding of percolation thresholds.

Findings

Existence of a truncation length N preserving infinite clusters for infinite expected degree.

Infinite clusters exist almost surely when expected degree exceeds 10^{400} in isotropic cases.

Results apply to long-range q-states Potts model, linking percolation and statistical physics.

Abstract

Consider independent long-range percolation on $\mathbb{Z}^d$ for $d\geq 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\geq 3$, we show that if the expected degree of the origin is at least $10^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model.