  In this work we study a natural transition mechanism describing the passage
from a quenched (almost sure) regime to an annealed (in average) one, for a
symmetric simple random walk on random obstacles on sites having an identical
and independent law. The transition mechanism we study was first proposed in
the context of sums of identical independent random exponents by Ben Arous,
Bogachev and Molchanov in [Probab. Theory Related Fields 132 (2005) 579--612].
Let $p(x,t)$ be the survival probability at time $t$ of the random walk,
starting from site $x$, and let $L(t)$ be some increasing function of time. We
show that the empirical average of $p(x,t)$ over a box of side $L(t)$ has
different asymptotic behaviors depending on $L(t)$. T here are constants
$0<\gamma_1<\gamma_2$ such that if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with
$\gamma>\gamma_1$, a law of large numbers is satisfied and the empirical
survival probability decreases like the annealed one; if $L(t)\ge e^{\gamma
t^{d/(d+2)}}$, with $\gamma>\gamma_2$, also a central limit theorem is
satisfied. If ${L(t)\ll t}$, the averaged survival probability decreases like
the quenched survival probability. If $t\ll L(t)$ and $\log L(t)\ll
t^{d/(d+2)}$ we obtain an intermediate regime. Furthermore, when the dimension
$d=1$ it is possible to describe the fluctuations of the averaged survival
probability when $L(t)=e^{\gamma t^{d/(d+2)}}$ with $\gamma<\gamma_2$: it is
shown that they are infinitely divisible laws with a L\'{e}vy spectral function
which explodes when $x\to0$ as stable laws of characteristic exponent
$\alpha<2$. These results show that the quenched and annealed survival
probabilities correspond to a low- and high-temperature behavior of a
mean-field type phase transition mechanism.
