The strong path localization conjecture for directed polymers

Let $u$ be the law of the random environment, and let $\beta$ denote the inverse temperature. For a polymer measure $\mu_{\beta,n}$ on paths of length $n$, let $\ell(\mu_{\beta,n})$ and $\rho(\mu_{\beta,n})$ denote the quenched localization observables defined in the paper. Under suitable conditions on $u$, Strong path localization conjecture. There \exists $\beta_0$ such that for every $\beta\geq\beta_0$, there \exists $\delta>0$ such that

$$\lim_{n\to\infty}\mathbb{P}(\ell(\mu_{\beta,n})\geq\delta)=\lim_{n\to\infty}\mathbb{P}(\rho(\mu_{\beta,n})\geq\delta)=1.$$

This asserts a stronger form of quenched path localization than the main theorem, which gives an upper bound on the probability that the localization observables are small for bounded-support environments satisfying the stated regularity assumptions. The conjecture concerns their asymptotic behavior as the polymer length tends to infinity.