Persistence in the zero-temperature dynamics of the $Q$-states Potts model on undirected-directed Barabási-Albert networks and Erdös-Rényi random graphs

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} Barabási-Albert networks and Erdös-Rényi random graphs. In this model it is found that $P(t)$ decays exponentially to zero in short times for {\it directed} and {\it undirected} Erdös-Rényi random graphs. For {\it directed} and {\it undirected} Barabási-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.

arXiv comment14 pages, 8 figures for IJMC

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