Logarithmic-scale speed at the zero-speed boundary

Characterize the phase transition in the logarithmic-scale speed \(\lim_{n\to\infty}\log |X_n|/\log n\) for a transient random walk in a random environment on \(\mathbb{Z}^+\) when the limiting harmonic parameter satisfies \(s_w=1\), particularly at the environmental exponent value \(\alpha=1\), under Conditions A and B.

Background

For the half-line $\mathbb{Z}^+$, the paper establishes the ballistic speed formula when $s_w<1$ under a regularity condition and explains that the walk can be transient but have zero speed when $s_w\geq 1$. A multiplicative-cascade example yields the borderline case $s_w=1$, motivating a conjecture about the logarithmic growth rate of the walk. The proposed transition is expected to occur primarily at the value $\alpha=1$.