Quenched localization conjecture for the rescaled running supremum

Let $\omega$ be the random environment, let $\alpha>1$, and let $\overline X_t$ denote the rescaled running supremum of the Mott random walk at time $t$. Write $P^{\alpha,0}_\omega$ for the quenched law of the walk, conditional on the environment. An environment-measurable process $(\Gamma_t)_{t\geq 0}$ is a process of subsets of localization sites in $\omega$.

Quenched localization conjecture. There exists an environment-measurable process $(\Gamma_t)_{t\geq 0}$ with $\Gamma_t\subseteq\omega$ and

$$|\Gamma_t|=2+\left\lfloor\frac{1}{\alpha-1}\right\rfloor$$

such that

$$\lim_{t\to\infty}P^{\alpha,0}_\omega\left(\overline X_t\in\Gamma_t\right)=1,\qquad \mathbf{P}\text{-a.s.}$$

Moreover, no environment-measurable set satisfying this localization property has smaller cardinality than $\Gamma_t$. This conjecture concerns the possible asymptotically visible delays caused by near-maximal resistance barriers between successive localization sites; the preceding results show that intermediate values are visited for a vanishingly small time, but do not rule out such delays.