Active-absorbing phase transition and small world behaviour in Ising model on finite addition type networks in two dimensions

TL;DR

This paper studies the phase transitions and small world properties of the Ising model on a two-dimensional lattice with added random bonds, revealing a rich phase diagram with distinct active and absorbing states.

Contribution

It introduces a novel network model with controlled shortcuts and maps the phase diagram of the Ising model on this network, highlighting the interplay between network structure and phase behavior.

Findings

Small world behavior occurs along a specific line in parameter space.

The phase diagram shows regions with active and absorbing states.

Different types of phase transitions are identified in the model.

Abstract

We consider the ordering dynamics of the Ising model on a square lattice where an additional fixed number of bonds connect any two sites chosen randomly. The total number of shortcuts added is controlled by two parameters $p$ and $\alpha$. The structural properties of the network are investigated which show that that the small world behaviour is obtained along the line $\alpha=\frac{\ln (N/2p)}{\ln N}$, which separates regions with ultra small world like behaviour and short ranged lattice like behaviour. We obtain a rich phase diagram in the $p-\alpha$ plane showing the existence of different types of active and absorbing states to which Ising model evolves and their boundaries.