  We consider a model of a polymer in $\mathbb{Z}^{d+1}$, 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 $N\to\infty$, when $d\geq3$ 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.
