and the necessary notation. After stating our results we offer an informal discussion, references, and describe a conjecture which makes our results a part of a more general picture.

Model description and main results
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Let $V(x,\omega),\ x\in{\mathbb{Z}}$, be i.i.d. random variables on a probability space $(\Omega,{{\mathcal}F},{\mathbb{P}})$ such that $$\label{pot}
  {\mathbb{P}}(V(0,\cdot)\ge 0)=1,\quad {\mathbb{P}}(V(0,\cdot)>0)>0,\quad\text{and}\quad {\mathbb{E}}[V(0,\cdot)]<\infty.$$ These random variables represent a random potential on ${\mathbb{Z}}$. Given a realization of the potential, i.e. for each fixed $\omega\in\Omega$, we consider a Markov chain on ${\mathbb{Z}}\cup\{\dagger\}$ with transition probabilities: $$\label{tp}
p(x,y,\omega)
  =
  \begin{cases}
    1-e^{-V(x,\omega)},&\text{if } y=\dagger,\ x\ne\dagger;\\
    \dfrac12\, e^{-V(x,\omega)},&\text{if } y=x\pm 1,\ x\ne\dagger;\\
    1,&\text{if } y=\dagger,\ x=\dagger;\\
    0,&\text{otherwise}.\\
  \end{cases}$$ Informally, this is a so called “killed random walk”: at each site $x$ the walk either gets killed with probability $1-e^{-V(x,\omega)}$ (and moves to the absorbing state $\dagger$) or survives and moves to one of the two neighboring sites with equal probabilities $e^{-V(x,\omega)}/2$. The corresponding path measure and the expectation with respect to it will be denoted by ${P}^\omega$ and ${E}^\omega$ respectively. Unless stated otherwise, the killed random walk starts from $0$.

Denote by $(S_n)_{n\ge0}$ a path on ${\mathbb{Z}}\cup\{\dagger\}$ and set $$\label{taux}
  \tau_x:=\inf\{n>0:\,S_n=x\}\in {\mathbb{N}}\cup\{\infty\}.$$ We shall put different measures on the set of nearest neighbor paths. 