Log-gamma polymer convergence to the Airy process

Fix $\theta,r>0$ and let $\tilde{f}_N^{LG}$ be the random continuous function from Definition 2. Then, as $N\to\infty$, the random functions $\tilde{f}_N^{LG}$ converge weakly in $(C(\mathbb{R}),\mathcal{C})$ to

$$\mathcal{L}^{Airy}_1(x)=2^{-1/2}(\mathcal{A}(x)-x^2),$$

where $\mathcal{A}$ is the Airy$_2$ process. Log-gamma polymer Airy conjecture. The weak convergence above holds for every fixed $\theta,r>0$. The Airy$_2$ process is a random continuous process, and this conjecture predicts the KPZ scaling limit of the log-gamma polymer under the appropriate $3:2:1$ scaling; the statement remains unproved in the source.