10 problems

Let $(T)$, $(T')$, and $(T)_\gamma$ for $\gamma\in(0,1)$ be Sznitman's ballisticity conditions for a uniformly elliptic, independent and identically distributed random walk in rand…

The transience-condition $(T)$ conjecture. Directional transience along an open set $\mathcal U_{\ell}$ is equivalent to condition $(T)|\ell$.

The strong ballistic conjecture. Any $d$-dimensional random walk in an i.i.d. uniformly elliptic random environment that is directionally transient along an open set of directions…

Consider a random walk in Dirichlet environment (RWDE) with parameter vector $(\alpha_i)_{i=1,\ldots,2d}$ in dimension $d\ge 2$, and let … Write $(T)'$ for the strengthened ballist…

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…

Let $d\geq 2$, and consider a random walk in an independent and identically distributed uniformly elliptic random environment on $\mathbb{Z}^d$. Let $U\subset\mathbb{S}^{d-1}$ be a…

For a random walk in random environment on $\mathbb{Z}^d$, let Sznitman's (T') condition denote the ballisticity condition introduced in the cited work, and let ballisticity mean t…

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…

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…