Umbrales de percolación de sitios. Pequeñas celdas bidimensionales asimétricas

W. Lebrecht, J.F. Valdés

Abstract


Site percolation thresholds $p_c$ and critical exponent $\nu$
associated to square lattices, triangular lattices and hexagonal
lattices are obtained. We consider a methodology consisting in the
growth in size of cells for each geometry, denoted for $M$. A site
is occupied with probability $p$ and $1-p$ if it is not occupied.
Two directions of the plane: horizontal and vertical, through
asymmetrical cells are considered for studying site percolation
phenomena, so, a percolation functions associated to horizontal or
vertical direction, $f^H(M,p)$ or $f^V(M,p)$ are obtained
respectively. Using finite scaling techniques, the critical points
at the thermodynamic limit are obtained. Site percolation
thresholds are compared through three different ways: first, using
the maximum of the derivative of the function $f^{(H,V)}(M,p)$
denoted by $p_p^{(H,V)}(M)$, second, considering the solution of
the equation $f^{(H,V)}(M,p)=p$, denoted by $p_g^{(H,V)}(M)$, and
third, using the cross-point of the curves associated to
percolation thresholds for horizontal and vertical directions,
represented by $p_f(M)$. Critical exponent $\nu$ is obtained
through two different ways: first, using the maximum of the
derivative defined as $f'^{(H,V)}(M,p_p)$, and second, considering
the cross point of both derivatives $f'(M,p_f)$. The values
associated to site percolation thresholds and critical exponent
$\nu$ are in good agreement with the similar ones informed in
literature, validating the methodology proposed here.

Keywords


Percolation; percolation threshold; critical exponent

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Revista Mexicana de Física

ISSN: 2683-2224 (on line), 0035-001X (print)

Bimonthly publication of Sociedad Mexicana de Física, A.C.
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