# Abstract

Crossing Random Walks and Stretched Polymers at Weak Disorder
D. Ioffe, Y. Velenik
Ann. Probab.
40,
714-742
(2012).
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 non-negative random environment. Alternatively, the model describes crossing random walks in a random potential, see Chapter~5 of Sznitman's book for the original Brownian motion formulation.

It was recently shown by Flury and by Zygouras that, in such a setting, the quenched and annealed free energies coincide in the limit $N\to\infty$, when $d\geq 3$ 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.

It was recently shown by Flury and by Zygouras that, in such a setting, the quenched and annealed free energies coincide in the limit $N\to\infty$, when $d\geq 3$ 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.

**Key words:**Polymers, crossing random walk, weak quenched disorder, diffusivity, Ornstein-Zernike theory**Files:**PDF file, Published version, bibtex