# Biased random walk conditioned on survival among Bernoulli obstacles: subcritical phase

## TL;DR

This paper studies a biased random walk avoiding obstacles on a lattice, revealing its confinement scale in the sub-ballistic phase and establishing large deviation principles for its endpoint distribution, extending previous results.

## Contribution

It proves the confinement scale of the walk in the sub-ballistic phase and establishes large deviation principles for the endpoint distribution at intermediate scales.

## Key findings

- Random walk is contained in a ball of radius O(N^{1/(d+2)}) in the sub-ballistic phase.
- Large deviation principles are established for the endpoint distribution at scales between N^{1/(d+2)} and o(N^{d/(d+2)}).
- Results extend and improve earlier work by Sznitman.

## Abstract

We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on ${\mathbb Z}^d$ ($d\geq 2$) up to time $N$. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. We prove that in the sub-ballistic phase, the random walk is contained in a ball of radius $O(N^{1/(d+2)})$, which is the same scale as for the unbiased case. As an intermediate step, we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between $N^{1/(d+2)}$ and $o(N^{d/(d+2)})$. These results improve and complement earlier work by Sznitman [Ann. Sci. Ecole Norm. Sup. (4), 28(3):345--370, 371--390, 1995].