Mathematics – Probability
Scientific paper
2010-02-23
Annals of Probability 2012, Vol. 40, No. 2, 714-742
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/10-AOP625 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/10-AOP625
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.
Ioffe Dmitry
Velenik Yvan
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