论文标题
关于RWRE的瞬态和弹道行为之间的联系
On the connection between transient and ballistic behaviours for RWRE
论文作者
论文摘要
我们研究了在随机环境（RWRE）中随机行走的弹道猜想的强烈形式。该猜想断言，任何在非空的开放指示方向上是瞬态的RWRE都满足条件$（t）$（对于不太可能退出概率的退火指数衰减）。具体来说，我们引入了弹道条件，该条件在多项式条件大于$ d-1 $持有的情况下就可以实现。在该假设下，我们证明了条件$（t）$，这将这种情况变成了最弱的弹道性假设。我们记得，标准论证证明弹道性条件意味着定向瞬态至少需要多项式衰减，大于$ d $。此外，在一维情况下，我们提供了另一种证据，证明了瞬态行为与不太可能退出概率退火任意衰减之间的等效性，我们希望该新参数可以在更高的维度中使用。
We study the strong form of the ballistic conjecture for random walks in random environments (RWRE). This conjecture asserts that any RWRE which is directionally transient for a nonempty open set of directions satisfies condition $(T)$ (annealed exponential decay for the unlikely exit probability). Specifically, we introduce a ballisticity condition which is fulfilled as soon as a polynomial condition of degree greater than $d-1$ holds. Under that hypothesis we prove condition $(T)$, which turns this condition into the weakest-known ballisticity assumption. We recall that standard arguments to prove that a ballisticity condition implies directional transience require at least polynomial decay greater than degree $d$. Furthermore, in the one dimensional case we provide an alternative proof which proves the equivalence between transient behaviour and annealed arbitrary decay for the unlikely exit probability, we expect that this new argument might be used in higher dimensions.
