CROSSING RANDOM WALKS AND STRETCHED POLYMERS AT WEAK DISORDER

We consider a model of a polymer in Z(d+1), constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246-280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528-1583; Probab. Theory Related Fields 143 (2009) 615-642] that, in such a setting, the quenched and annealed free energies coincide in the limit N -> infinity, when d >= 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.