Persistence in the zero-temperature dynamics of the $Q$-states Potts model on undirected-directed Barab\'asi-Albert networks and Erd\"os-R\'enyi random graphs

TL;DR

This study investigates the zero-temperature Glauber dynamics of the Potts model on different networks, revealing exponential decay of persistence and blocking phenomena depending on network type and parameters.

Contribution

It demonstrates that the persistence probability exhibits blocking on Barabási-Albert networks across a wide range of states, contrasting with decay on Erdős-Rényi graphs.

Findings

Persistence decays exponentially to zero on Erdős-Rényi graphs.

Persistence approaches a non-zero constant on Barabási-Albert networks, indicating blocking.

An exception occurs at very high Q in undirected networks, possibly due to finite-size effects.

Abstract

The zero-temperature Glauber dynamics is used to investigate the persistence probability $P(t)$ in the Potts model with $Q=3,4,5,7,9,12,24,64, 128$, $256, 512, 1024,4096,16384 $,..., $2^{30}$ states on {\it directed} and {\it undirected} Barab\'asi-Albert networks and Erd\"os-R\'enyi random graphs. In this model it is found that $P(t)$ decays exponentially to zero in short times for {\it directed} and {\it undirected} Erd\"os-R\'enyi random graphs. For {\it directed} and {\it undirected} Barab\'asi-Albert networks, in contrast it decays exponentially to a constant value for long times, i.e, $P(\infty)$ is different from zero for all $Q$ values (here studied) from $Q=3,4,5,..., 2^{30}$; this shows "blocking" for all these $Q$ values. Except that for $Q=2^{30}$ in the {\it undirected} case $P(t)$ tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.