Umbrales de percolación de sitios. Pequeñas celdas bidimensionales asimétricas

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.