On the moments of the meeting time of independent random walks in random environment

We consider, in the continuous time version, $\gamma$ independent random walks on $\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\gam$ be the first meeting time of one pair of the $\gamma$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_\gamma$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\Eo$ the quenched expectation, we show that, for almost all environments $\omega$, $\Eo[T_\gamma^{c}]$ is finite for $c<\gamma(\gamma-1)/2$ and infinite for $c>\gamma(\gamma-1)/2$.