3 problems

Let $d\geq 3$ and let $\beta_c,\bar \beta_c\in(0,\infty]$ be the critical points such that weak disorder holds for $\beta\in[0,\beta_c)$, strong disorder holds for $\beta>\beta_c$,…

Let $d\geq 3$ and let $\overline\kappa_{\rm cr}$ denote the critical jump rate separating the weak- and strong-disorder phases for the continuous-time directed polymer. Critical st…

Assume (Dir), (Ind), (Loc), $V\in\mathcal{L}$, and $\Lambda_q(V)<\infty$. Let $\underline h_\infty^\lambda$ and $\bar h_\infty^\lambda$ denote the lower and upper limiting free-ene…