14 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…

Greven–den Hollander conjecture. For every $x>v$,

Let $d\geq 2$. Consider a nearest-neighbor random walk in random environment (RWRE) on $\mathbb{Z}^d$ whose transition probabilities are uniformly bounded below by a positive const…

Non-zero area anomaly conjecture. Roughly speaking, any reasonably asymmetric RWRE from this ballistic class should have a non-zero area anomaly.

Let $d\geq 2$ and let $\mathbf{u}\in S^{d-1}$ be a direction. Write $A_{\mathbf{u}}$ for the event of transience in direction $\mathbf{u}$, and let $\tau_{\mathbf{u},1}$ be the fir…

Let $d\geq 2$. Consider a uniformly elliptic, ballistic, nearest-neighbor random walk in random environment, and let the regeneration times be defined in a direction of transience.…

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 $(X_n)$ be a one-dimensional random walk in a random environment satisfying the i.i.d. uniformly elliptic assumption, with $E_P\log((1-\omega_0)/\omega_0)<0$. Write…

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…