SCALING FUNCTION FOR THE NUMBER OF ALTERNATING PERCOLATION CLUSTERS ON SELF-DUAL FINITE SQUARE LATTICES

We consider bond percolation with a bond probability p on a L(1) X L(2 ) self-dual square lattice with periodic boundary conditions in the ho rizontal direction and free boundary conditions in the vertical direct ion, terminated at the top and bottom by a row of vertical and horizon tal bonds, respectively. We define the number M of alternating percola tion clusters as the minimum of n(p) and n(n), where n(p) is the numbe r of independent percolating dusters connecting sites on the top and b ottom edges, and n, is the number of percolating clusters in the compl ementary configuration on the dual lattice, a bond being present in th e complementary configuration if and only if it is absent in the origi nal configuration. We evaluate the probability W-M(a)(L(1),L(2),p) for finding a given value of M and find that, for a given aspect ratio L( 1), /L(2) all data of W-M(a)(L(1),L(2),p) near the critical point p(c) fall on the same scaling function F-M(a) which is symmetric with resp ect to the scaling variable for all M.