Effects of long-range interactions in the one-dimensional Sznajd model

The Sznajd model is a one-dimensional, binary, voter-like model used to study consensus in systems where information flows outward from like-minded agent pairs. Here, we introduce long-range interactions to the Sznajd model, quantified by the parameter p in analogy with the dynamic and static small-world rewiring parameter (p$\to $1 is the mean-field limit, p$\to $0 is the 1-D limit). We use Monte Carlo simulations and finite-size scaling analyses to characterize the exit probability for p$\ne $0, finding a step function that depends on two p-dependent exponents. By examining the p$\to $0 limit of these exponents, we comment on the functional form of the exit probability in one dimension, which has been an open question. We complement this limiting approach (letting p$\to $0, which offers considerable computational speedup over the pure p$=$0 case) by also simulating the p$=$0 case via a parallel algorithm. This investigation also probes the dependence of consensus time and system magnetization on p.