REPLICA MODEL AT LOW INTEGER-N FOR DIRECTED POLYMERS IN (1+1) DIMENSIONS

We study directed polymers in a (1 + 1)-dimensional disordered environ ment with discrete space and time. For fixed L (polymer length) and r (disorder parameter) the quantity (Z(n)) has two crossovers in n, one related to the radius of convergence of the cumulant expansion for lnZ and one related to the discreteness of the lattice. The existence of the first crossover [at n1(L)-->0 as L --> infinity] weakens the custo mary argument relating the behavior of the cumulants of lnZ to the n d ependence of E(n) = -lim(L-->infinity)ln[Z(n)]/L. The second crossover [at n2(r)--> infinity as r --> 0] is explored here by computing E2 an alytically and E3 numerically to high accuracy for several values of r in two models.