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A Proof of Sznitman's Conjecture about Ballistic RWRE
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2019-11-27 , DOI: 10.1002/cpa.21877
Enrique Guerra 1 , Alejandro F. Ramı́rez 1
Affiliation  

We consider a random walk in a uniformly elliptic i.i.d. random environment in $\mathbb Z^d$ for $d\ge 2$. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in 2001 and 2002, Sznitman defined the so called conditions $(T)$ and $(T')$. The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width $L$. The second one is the requirement that for all $\gamma\in (0,1)$ condition $(T)_\gamma$ is satisfied, which in turn is defined as the requirement that the decay is like $e^{-CL^\gamma}$ for some $C>0$. In this article we prove a conjecture of Sznitman of 2002, stating that $(T)$ and $(T')$ are equivalent. Hence, this closes the circle proving the equivalence of conditions $(T)$, $(T')$ and $(T)_\gamma$ for some $\gamma\in (0,1)$ as conjectured by Sznitman, and also of each of these ballisticity conditions with the polynomial condition $(P)_M$ for $M\ge 15d+5$ introduced by Berger, Drewitz and Ramirez in 2014.

中文翻译:

Sznitman 关于弹道 RWRE 猜想的证明

我们考虑在 $\mathbb Z^d$ 中 $d\ge 2$ 的均匀椭圆 iid 随机环境中的随机游走。人们相信,只要随机游走在给定方向上是短暂的,它就必然是弹道的。为了量化证明这种等效性所需的差距,引入了几个弹道条件。特别是在 2001 年和 2002 年,Sznitman 定义了所谓的条件 $(T)$ 和 $(T')$。第一个是要求一组平板的某些不太可能的退出概率以其宽度 $L$ 呈指数快速衰减。第二个是要求对于所有 $\gamma\in (0,1)$ 条件 $(T)_\gamma$ 都满足,这又被定义为要求衰减类似于 $e^{- CL^\gamma}$ 对于某些 $C>0$。在本文中,我们证明了 2002 年 Sznitman 的一个猜想,声明 $(T)$ 和 $(T')$ 是等价的。因此,这关闭了证明 Sznitman 推测的某些 $\gamma\in (0,1)$ 条件 $(T)$、$(T')$ 和 $(T)_\gamma$ 的等价性的圆,以及这些弹道条件中的每一个,以及由 Berger、Drewitz 和 Ramirez 在 2014 年引入的多项式条件 $(P)_M$ for $M\ge 15d+5$。
更新日期:2019-11-27
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