  Place an obstacle with probability $1-p$ independently at each vertex of
$\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon
hitting one of the obstacles. For $d \geq 2$ and $p$ strictly above the
critical threshold for site percolation, we condition on the environment such
that the origin is contained in an infinite connected component free of
obstacles. It has previously been shown that with high probability, the random
walk conditioned on survival up to time $n$ will be localized in a ball of
volume asymptotically $d\log_{1/p}n$. In this work, we prove that this ball is
free of obstacles, and we derive the limiting one-time distributions of the
random walk conditioned on survival. Our proof is based on obstacle
modifications and estimates on how such modifications affect the probability of
the obstacle configurations as well as their associated Dirichlet eigenvalues,
which is of independent interest.
