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

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 $,\mathrm{.}\mathrm{.}\mathrm{.},$2^{30}$ 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, $P(\mathrm{\infty })$ is different from zero for all $Q$ values (here studied) from $Q=3,4,5,…, 2^{30}$;thisshows\mathrm{"}blocking\mathrm{"}forallthese$Q$values\mathrm{.}Exceptthatfor$Q=2^{30}$intheundirectedcase$P(t)$ 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.