In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/rd+σ, 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 σ the critical value Cc at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10-3, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for Cc are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the expansion used with the introduction of a suitably defined effective dimension deff relating the long-range model with a short-range one in dimension deff. 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 σ>0.

One-dimensional long-range percolation: A numerical study / Gori, G.; Michelangeli, Marco; Defenu, N.; Trombettoni, A.. - In: PHYSICAL REVIEW. E. - ISSN 2470-0045. - 96:1(2017), pp. 1-9. [10.1103/PhysRevE.96.012108]

One-dimensional long-range percolation: A numerical study

Gori, G.;Michelangeli, Marco;Defenu, N.;Trombettoni, A.
2017-01-01

Abstract

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C/rd+σ, 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 σ the critical value Cc at which percolation occurs. The critical exponents in the range 0<σ<1 are reported. Our analysis is in agreement, up to a numerical precision ≈10-3, with the mean-field result for the anomalous dimension η=2-σ, showing that there is no correction to η due to correlation effects. The obtained values for Cc are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ=1 and from the expansion used with the introduction of a suitably defined effective dimension deff relating the long-range model with a short-range one in dimension deff. 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 σ>0.
2017
96
1
1
9
012108
http://harvest.aps.org/bagit/articles/10.1103/PhysRevE.96.012108/apsxml
https://arxiv.org/abs/1610.00200
Gori, G.; Michelangeli, Marco; Defenu, N.; Trombettoni, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/70884
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