Square lattice directed paths adsorbing on the lineY = qX

Abstract

Consider fully directed paths from the origin in the square lattice and confined to theq/p-wedge formedby the Y-axisand the line Y = (q/p)X, where (p, q) is a pair of positive integers. If such a path is constrained to end in a vertex in the lineY = (q/p)X, then we call ita q/p-Dyck path.Suppose that gq/p(t, z) is thegenerating function of q/p-Dyck paths, and suppose that the generating variable for edges (or steps) ist, andthat z is the generating variable of vertices in the lineY = (q/p)X (these are visits to theadsorbing line Y = (q/p)X). In thispaper it is shown that q/1-Dyck or 1/q-Dyck paths adsorb at the critical value ofz given byzq/1 = z1/q = q + 1. Evidencethat zq/p ≥ p+q is presented and the results are extended to models of directed paths from the origin in ther-wedge formedby the Y-axisand the line Y = rX, where r ≥ 0 is areal number. If gr(t) is the generating function of such paths, then let the radius of convergence ofgr(t) betr. I find a generalexpression for tr,and prove that if r = π, then tπ = ππ/(1 + π)/(1 + π). This result estimates the exponential rate of growth of the number of directed paths in aπ-wedge.Models of q/p-Dyck paths with an area generating variable are also considered, andthe generating function of this model is determined in the case wherep = 1.