Publication

The Extinction of the Contact Process in a One-Dimensional Random Environment With Long-Range Interactions

2026

Abstract

We study the contact process on the long-range percolation cluster on (Formula presented.) where each edge (Formula presented.) is open with probability (Formula presented.) for (Formula presented.). Using a renormalization procedure, we apply the Peierls-type argument to prove that the contact process dies out if the transmission rate is lower than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in Hilário et al.

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2026

Abstract

Official source

Related concepts (29)

Percolation threshold

The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them.

Percolation critical exponents

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered.

Percolation theory

In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology).