One-dimensional long-range percolation: A numerical study

Year: 2017

Authors: Gori G., Michelangeli M., Defenu N., Trombettoni A.

Autors Affiliation: CNR IOM DEMOCRITOS Simulat Ctr, Via Bonomea 265, I-34136 Trieste, Italy; SISSA, Via Bonomea 265, I-34136 Trieste, Italy; INFN, Sez Trieste, Via Bonomea 265, I-34136 Trieste, Italy.

Abstract: In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/r(d+sigma), where r is the distance length between distinct sites and d = 1. We introduce and test an order-N Monte Carlo algorithm and we determine as a function of sigma the critical value C-c at which percolation occurs. The critical exponents in the range 0 < sigma < 1 are reported. Our analysis is in agreement, up to a numerical precision approximate to 10(-3), with the mean-field result for the anomalous dimension eta = 2 -sigma, showing that there is no correction to eta due to correlation effects. The obtained values for C-c are compared with a known exact bound, while the critical exponent. is compared with results from mean-field theory, from an expansion around the point sigma = 1 and from the e-expansion used with the introduction of a suitably defined effective dimension deff relating the long-range model with a short-range one in dimension d(eff). We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with sigma > 0.

Journal/Review: PHYSICAL REVIEW E

Volume: 96 (1)      Pages from: 12108-1  to: 12108-9

More Information: We gratefully acknowledge discussions with Miguel Ibanez Berganza, Luca Lepori, Stefano Ruffo, Slava Rychkov, and Hirohiko Shimada. Useful correspondence with Peter Grassberger is also acknowledged. Hospitality in the Program Conformal Field Theories and Renormalization Group Flows in Dimensions d > 2 at the Galileo Galilei Institute for Theoretical Physics, Florence (Italy), where part of this work was performed, is gratefully acknowledged. G.G. and A.T. acknowledge funding support from Progetto Premiale Anno 2012 ABNANOTECH-Atom-based technology.
KeyWords: Ising-model; Critical Exponents; Phase-transitions; Critical-behavior; Clusters; Systems; Order
DOI: 10.1103/PhysRevE.96.012108

Citations: 29
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