4 problems

Let $R_5:=\{(\gamma_c,\gamma_a): \gamma_a>1/2,\ \max\{0,2/\gamma_a-1,(\gamma_a-5)/(\gamma_a-2)\}<\gamma_c<3/(2\gamma_a)\}$. Consider the directed-polymer partition function…

Let $\hat{\beta}>0$ be finite, and let $\hat{I}_{{\hat{\beta}},\infty}$ be the infinite-volume variational set associated with the model. Finite-point conjecture. We conjecture tha…

Let $\gamma=0$, and let the renewal law satisfy $K(n):=\mathrm{P}(\tau_1=n)\sim L(n)n^{-\rho}$ with $\rho>1$. Let $\mathrm{P}_{{\beta}_N,h,N}^{{\omega}}$ denote the corresponding p…

Let $\tau$ be the renewal process and let $N$ be the system size, so that $\tau/N$ denotes its rescaled renewal set. In the regime $\gamma\geq 1$, the entropy optimization yields t…