Strong Disorder and Polymer Superdiffusivity

Definitions and mathematical statement

The supplied assessment identifies the short question "Does strong disorder imply superdiffusivity?" as concerning directed polymers in a random environment. It does not specify the model or the precise transverse-fluctuation criterion.

In the standard discrete example, let $(S_j)$ be simple random walk on ${\mathbb{Z}}^d$, $d\geq1$, starting at zero, let $(\omega_{j,x})$ be independent identically distributed environmental variables, and assume $\lambda(\beta)=\log\mathbb E[e^{\beta\omega_{1,0}}]<\infty$ for the chosen $\beta>0$. With $E_0$ denoting random-walk expectation and $\mathbb E$ environmental expectation, define \[ Z_n^\omega=E_0\exp\!\left(\beta\sum_{j=1}^n\omega_{j,S_j}\right),\qquad W_n^\omega=e^{-n\lambda(\beta)}Z_n^\omega, \] \[ \frac{dQ_{n,\beta}^\omega}{dP_0} =\frac{\exp(\beta\sum_{j=1}^n\omega_{j,S_j})}{Z_n^\omega}. \] The normalized partition functions form a nonnegative mean-one martingale in the time environment. In this convention strong disorder is the assertion $W_\infty=0$ almost surely. It should not be silently replaced by the stronger strict quenched/annealed free-energy-gap condition often called very strong disorder.

The desired implication is from this strong-disorder condition to transverse spreading faster than the underlying diffusive scale $n^{1/2}$, under the intended polymer law. For example, one possible power-law second-moment criterion would be \[ \liminf_{n\to\infty} \frac{\log Q_{n,\beta}^\omega[|S_n|^2]}{2\log n}>\frac12 \quad\text{almost surely in the environment}. \] This is not the unique definition of superdiffusivity: divergence of the second moment divided by $n$, escape from every fixed diffusive ball, maximal path displacement, and an annealed rather than quenched exponent are different formulations. Centered thermal variance also differs from uncentered displacement in a random environment. The source selects none, so the display is an explicitly conditional interpretation rather than a claim of its exact intended conclusion. The nearest-neighbor, independent-environment and finite-exponential-moment assumptions likewise describe this example, not an extracted universal model class.

The supplied literature triage checked 21 August 2026 reports known localization and strong-disorder results but no established general implication to superdiffusivity. Localization, weak-disorder diffusivity, and transverse superdiffusivity are distinct statements; the cited progress is not treated as proving the requested implication or its negation.

Short English statement

For directed polymers, must vanishing of the normalized partition-function martingale force spreading faster than the diffusive scale? The precise polymer model and quenched or annealed meaning of superdiffusivity are missing. The supplied assessment records localization results but no general implication.

Sources

• Local UnsolvedMath registry, permanent ID 30000126, exact source code catalogue entry; Strong Disorder and Polymer Superdiffusivity. Proposed year 2004, status open; corrected_verified short question.
• Supplied report citation: Large Scale Stochastic Dynamics (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/43, pp. 2229--2310, https://doi.org/10.4171/owr/2004/43. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "Comets--Yoshida, diffusivity at weak disorder", https://arxiv.org/abs/math/0411223; "Carmona--Hu, strong disorder and localization", https://arxiv.org/abs/math/0601670; "Directed-polymer phase-transition review", https://doi.org/10.1016/j.spa.2024.104431. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.