Crossover from mean-field to $2d$ Directed Percolation in the contact process

TL;DR

This study investigates how the contact process on spatially embedded networks transitions from mean-field behavior to 2D directed percolation as the range of long-range connections varies, revealing a crossover influenced by network topology.

Contribution

It demonstrates the crossover from mean-field to 2D directed percolation universality class in the contact process on networks with tunable long-range connections.

Findings

Crossover occurs for 3<α<4 as long-range connection probability varies.

Monte Carlo simulations confirm the universality class transition.

Finite-size scaling analysis supports the universality change.

Abstract

We study the contact process on spatially embedded networks, consisting of a regular square lattice with long-range connections. To generate the networks, a long-range connection is randomly added to each node $i$ of a square lattice, following the probability, $P_{ij}\sim{r_{ij}^{-\alpha}}$ , where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $\alpha$ is a tunable parameter. Extensive Monte Carlo simulations and a finite-size scaling analysis for different values of $\alpha$ reveal a crossover from the mean-field to $2d$ Directed Percolation universality class with increasing $\alpha$, in the range $3<\alpha<4$.