Crossing random walks and stretched polymers at weak disorder (Q414285)

A stretched polymer is represented by the set of nearest-neighbor paths \(\gamma=\{\gamma(0),\dots,\gamma(n)\}\) on \(\mathbb{Z}^{d+1}\) from the origin \(x=0\) to the hyperplane \(x^\mid=N\) -- here, \(x\) is decomposed into transverse and longitudinal parts, i.e., \(x=(x^\perp,x^\mid)\) with \(x^\perp \in \mathbb{Z}^{d}\), \(x^\mid \in \mathbb{Z}\). The polymer is subject to a quenched nonnegative random environment \(\{V^\omega(x)\}_{x \in \mathbb{Z}^{d+1}}\), \(\omega \in \Omega\), assumed to be i.i.d. The weight of a polymer is given by NEWLINE\[NEWLINE W^\omega_{\lambda,\beta}=\exp\{-\lambda n -\beta \sum_{l=1}^n V^\omega(\gamma(l))\}. NEWLINE\]NEWLINE Corresponding quenched and annealed partition functions are defined as \(D^\omega_N=\sum_\gamma W^\omega_{\lambda,\beta}(\gamma)\) and \(E D^\omega_N\). It is first proved that, under some 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 as \(N\to \infty\). This result improves the results in [\textit{M. Flury}, Ann. Probab. 36, No. 4, 1528--1583 (2008; Zbl 1156.60076); \textit{N.Zygouras}, Probab. Theory Relat. Fields 143, No. 3--4, 615--642 (2009; Zbl 1163.60050)].NEWLINENEWLINEThe second result describes the distribution of the transverse component \(\pi^\perp(\gamma)\) of \(\gamma(n)=(\pi^\perp(\gamma),\pi^\mid(\gamma))\). In the considered case, the polymer obeys a diffusive scaling, with the same diffusivity constant as in the annealed model.NEWLINENEWLINEAlternatively, the considered model describes crossing random walks in a random potential (see [\textit{M. P. W. Zerner}, Ann. Appl. Probab. 8, No. 1, 246--280 (1998; Zbl 0938.60098)]).