Geometry of the random walk range conditioned on survival among Bernoulli obstacles

We consider a discrete time simple symmetric random walk among Bernoulli obstacles on $mathbb{Z}^d$, $dgeq 2$, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time $N$, the random walk range is asymptotically contained in a ball of radius $varrho_N=C N^{1/(d+2)}$ for any $dgeq 2$. For $d=2$, it is also known that the range asymptotically contains a ball of radius $(1-epsilon)varrho_N$ for any $epsilon>0$, while the case $dgeq 3$ remains open. We complete the picture by showing that for any $dgeq 2$, the random walk range asymptotically contains a ball of radius $varrho_N-varrho_N^epsilon$ for some $epsilon in (0,1)$. Furthermore, we show that its boundary is of size at most $varrho_N^{d-1}(log varrho_N)^a$ for some $a>0$.