SERIES AND MONTE-CARLO STUDIES OF 2 AND 3 DIMENSIONS FOR AXIAL HYPERSCALING IN DIRECTED PERCOLATION

The relationship between the exponent gamma0, characterizing the divergence, at criticality, of the cluster mass measured on the anisotropic axis of directed percolation, and the susceptibility exponent gamma, while well established in two dimensions, both by series and simulation work, has been doubtful for a decade for three and higher dimensions. In the present paper we furnish extensive simulations for the body centered cubic site problem that focus on that quantity, and study further the ascending moments of the number of sites on the axis. These numerical results are highly compatible with the scaling relation gamma0 = gamma - Dnu(perpendicular-to), where D = 2 is the transverse dimensionality. They also indicate the existence of a fixed gap exponent for the size moments of this quantity. Extensive series expansions in two dimensions are in acceptable agreement with the same picture (with D = 1, above). While direct determination of gamma0 is still not satisfactory for some of the highly connected three-dimensional lattices we have adopted, the existence of a fixed gap exponent is confirmed in fairly reasonable agreement with the concurrent evidence of the BCC Monte Carlo studies.