The critical inverse-temperature characterization of path localization

Let $\kappa$ be the diffusivity parameter, let $\beta$ be the inverse-temperature parameter, and let $\Upsilon_c$ denote the critical value separating the localized and nonlocalized regimes. Define the critical set of inverse temperatures by

$${\mathcal N}\!{\mathcal C}_\kappa=\big\{\beta: |\beta|\geq (\kappa \Upsilon_c)^{1/2}\big\}.$$

Path-localization conjecture. The set of inverse temperatures for which path localization holds is

$${\mathcal N}\!{\mathcal C}_\kappa=\big\{\beta: |\beta|\geq (\kappa \Upsilon_c)^{1/2}\big\}.$$

The authors note that this conjecture holds for polymer models on trees; its status for the Anderson polymer model considered here is not established.