High-temperature Poisson variational limit conjecture for directed polymers in region $R_5$

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 $Z_{n,\beta_n}^{\omega}$ with disorder parameter $\gamma_a$, temperature parameter $\beta_n$, and disorder satisfying~. Let $\mathcal{P}=\{(x_i,t_i,u_i):i\geqslant1\}$ be a Poisson point process on $\mathds{R}\times[0,1]\times[0,\infty)$ with intensity one, and define

$$T:=\sup_{A\subset\mathcal{P}:|A|<\infty}\sum_{i\in A}\left(u_i^{-1/\gamma_a}-\frac{|x_i-x_{i-1}|^2}{2|t_i-t_{i-1}|}\right),$$

where $A$ is ordered by increasing $t_i$. Poisson variational limit conjecture. For $\gamma_a\in(1/2,2]$, $T$ is well-defined. Moreover, if $(\gamma_a,\gamma_c)\in R_5$ with $\gamma_a<2$ and $\omega$ satisfies~, then there exist constants $a_n,h_n$ such that

$$h_n=n^{\frac{3\gamma_a/2-\gamma_c}{4-2/\gamma_a}+o(1)}$$

and

$$h_n^{-2}\bigl(\log Z_{n,\beta_n}^{\omega}-a_n\bigr)\xrightarrow[]{(d)}T,\qquad\text{as }n\to\infty.$$

The conjecture predicts a universal Poissonian variational description of the centered and rescaled logarithmic partition function in the heavy-tailed high-temperature regime. The preceding discussion establishes finiteness of $T$ almost surely for $\gamma_a\in(1/2,1)$; well-definedness for the full interval $(1/2,2]$ and the stated convergence remain open.