Equivalence of Sznitman Ballisticity Conditions

Definitions and mathematical statement

The supplied record concerns Sznitman's ballisticity conditions for a uniformly elliptic i.i.d. random walk in random environment. The following spells out their standard nearest-neighbour interpretation. The record itself does not specify the allowed steps or give the exit-probability definition of these conditions; these conventions require confirmation against the cited source.

Let $d\geq2$ and let $E_d=\{\pm e_1,\ldots,\pm e_d\}$, where $e_i$ are the coordinate vectors. An environment is a family of probability vectors $\omega(x,\cdot)$ on $E_d$, indexed by $x\in{\mathbb{Z}}^d$. Let $\mathbb P$ be an environment law under which these vectors are independent and identically distributed. Uniform ellipticity means that there exists $\kappa>0$ such that \[ \mathbb P\left(\omega(x,e)\geq\kappa\text{ for every }x\in{\mathbb{Z}}^d\text{ and }e\in E_d\right)=1. \] Given $\omega$, let $P_{0,\omega}$ be the law of the walk starting at $0$ with transition probabilities \[ P_{0,\omega}(X_{n+1}=x+e\mid X_n=x)=\omega(x,e). \] Write $P_0(A)=\int P_{0,\omega}(A)\,\mathbb P({\,\mathrm{d}}\omega)$ for its annealed law. For a unit vector $u\in S^{d-1}=\{v\in{\mathbb{R}}^d:\|v\|_2=1\}$, define \[ H_a^u=\inf\{n\geq0:X_n\cdot u\geq a\},\qquad \widetilde H_a^u=\inf\{n\geq0:X_n\cdot u\leq a\}, \] with $\inf\varnothing=\infty$.

For $\ell\in S^{d-1}$ and $\gamma\in(0,1)$, the conventional condition $(T)_\gamma$ relative to $\ell$ is that there exists a neighbourhood $U$ of $\ell$ in $S^{d-1}$ such that, for every $u\in U$ and every $b>0$, \[ \limsup_{L\to\infty}L^{-\gamma}\log P_0\!\left(\widetilde H_{-bL}^{u}<H_L^{u}\right)<0, \] where $\log0=-\infty$. Condition $(T')$ relative to $\ell$ means that $(T)_\alpha$ relative to $\ell$ holds for every $\alpha\in(0,1)$; the neighbourhood may depend on $\alpha$.

Conjecture, under this conventional interpretation. For every integer $d\geq2$, every uniformly elliptic i.i.d. environment law as above, every $\ell\in S^{d-1}$, and every $\gamma\in(0,1)$, \[ (T)_\gamma\text{ relative to }\ell \quad\Longleftrightarrow\quad (T')\text{ relative to }\ell. \] The source supplies this equivalence conjecture, but does not explicitly supply the nearest-neighbour model, neighbourhood quantifiers, or boundary conventions used in the expanded definitions above.

Short English statement

The conjecture asks whether, for every dimension $d\geq2$ and direction $\ell$, Sznitman's condition $(T)_\gamma$ for any one exponent $\gamma\in(0,1)$ implies the corresponding conditions for every exponent in $(0,1)$. The expanded nearest-neighbour definitions require source confirmation. The supplied record labels the problem partially solved.

Sources

• Local UnsolvedMath registry, permanent ID $30001323$, source problem code catalogue entry, title: Equivalence of Sznitman Ballisticity Conditions; collection: Oberwolfach Reports Open Problems. The supplied record lists proposed year 2009 and does not identify a proposer. The conditions are attributed to Sznitman; this does not identify the proposer of the conjecture.
• Supplied source citation: Scaling Limits in Models of Statistical Mechanics (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/38, pp. 2115--2178. The background describes these reports as documenting mathematical workshops and their open-problem discussions. Literature-source label: original Oberwolfach report 4081. https://doi.org/10.4171/owr/2009/38
• Supplied literature-source label: Drewitz-Ramirez ballisticity conditions. https://arxiv.org/abs/0903.4465
• Supplied literature-source label: higher-dimensional extension. https://arxiv.org/abs/1005.0376
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that equivalence was proved for gamma above a dimension threshold and for all gamma in dimensions at least four, while the full low-dimensional statement remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status field is corrected_verified; its accompanying verification note says that the equivalence conjecture is complete and that dimension, parameter, direction, and ballisticity-condition notation are normalized. These are supplied editorial labels, not verification performed for this draft.