8 problems

Consider a random walk in a uniformly elliptic random environment in dimension $d\ge 2$ and a direction $l\in\mathbb S^{d-1}$. For $\gamma\in(0,1]$, condition $(T)_\gamma|l$ requir…

Sznitman's conjecture. Condition $(T')$ is equivalent to ballisticity.

Sznitman's transience-ballisticity conjecture. If the walk is transient in direction $l$, then it is ballistic in direction $l$.

Transience–ballisticity conjecture. Under these assumptions, RWRE that is transient with respect to all directions in an open subset of $\mathbb S^{d-1}$ is necessarily ballistic.

Let $d\geq2$, let $\mathbb{P}$ be uniformly elliptic, and let $l\in\mathbb{S}^{d-1}$. Assume condition $(T')|l$ holds. For $\beta\in(0,1)$ and $L>0$, define the slab … Let…

Transience–ballisticity conjecture. Every uniformly elliptic RWRE in an i.i.d. environment that is transient in a given direction is ballistic in the same direction. This conjectur…

Let $d\ge 2$ and let $l\in\mathbb S^{d-1}$. A random walk in a uniformly elliptic i.i.d. random environment is transient in the direction $l$ if … $P_0$-almost surely, and it is ba…

Consider nearest-neighbor random walk in a random environment on $\mathbb{Z}^d$ with an i.i.d. uniformly elliptic environment. Let $I_q$ and $I_a$ denote the quenched and averaged…