Conditional and uniform quenched CLTs for one-dimensional random walks among random conductances

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{conditional} invariance principle for the random walk, under the condition that it remains positive until time $n$. As a corollary of this result, we study the effect of conditioning the random walk to exceed level $n$ before returning to 0 as $n\to \infty$. One of the main tools for proving these conditional limit laws is the \textit{uniform} quenched functional Central Limit Theorem, that states that the convergence is uniform with respect to the starting point, provided that the starting point is chosen in a certain interval around the origin.