Computer simulations of statistical models and dynamic complex systems

Abstract

These notes concern the material covered by the authors during 4 classes on the Escola Brasileira de Mecânica Estatística, University of São Paulo at São Carlos, February 2004. They are divided in almost independent sections, each one with a small introduction to the subject and emphasis on the computational strategy adopted.

Block voter model: Phase diagram and critical behavior

“…4) as a function of the noise parameter q, for several lattice sizes L. For sufficiently large systems, these curves intercept each other in a sin- gle point U * (q c ). The value of q where occurs the intersection equals the critical noise q c , which is not biased by any assumption about critical exponents since, by construction, the Binder's cumulant presents zero anomalous dimension [16,17]. In Fig.…”

Majority-vote on directed Erdős–Rényi random graphs

“…To determine estimates for the critical point q c , we calculate the Binder fourthorder magnetization cumulant U at different values of the noise q and several network sizes N. Finite size scaling predicts that for sufficiently large systems, these curves should have a unique intersection point U * [27]. The value of q where this crossing occurs is the value of the critical noise q c which is not biased by any assumptions about critical exponents, since by construction, the Binder cumulant presents zero anomalous dimension, therefore it respects the correct critical behavior of the system near q c [27,28].…”

Dynamical model for competing opinions

“…the configurations are reached without the need of the external tuning of any parameter, and the power law suggests the existence of critical behavior. The self organized criticality (SOC) has been discussed in different places [16][17][18][19][20] and has been observed in many different systems [20][21][22][23][24]. In our case, this suggests that, on rare occasions, consensus would spontaneously be reached.…”

“…This is called self-organized criticality (SOC) and has been discussed in many different places [12,[22][23][24][25][26]. It has also been observed in a large diversity of systems [12,[22][23][24][25][26][27][28][29][30]. The asymptotic value of the power law exponent of the size distribution shown above lies within the values obtained with the random site percolation (RSP) whose exponent ranges from 2.05 for two-dimensional lattices [13] or 2.186 ± 0.002 for three-dimensional lattices [31] up to 2.5 for the Bethe lattice [13].…”