Improved universality bounds for directed polymers in the intermediate disorder regime

We prove universality of Tracy-Widom GUE fluctuations for directed polymers in $1+1$ dimensions in the intermediate disorder regime. Building on the Lindeberg replacement method of arXiv:2304.04871, we refine estimates for the measure of steep paths using probabilistic arguments. Our result extends the admissible range of the inverse temperature scaling parameter $\beta = n^{-\alpha}$ to all $\alpha \in (2/(3K+11), 1/4)$, provided the weights match $K$ moments with the log-Gamma distribution. For a general class of distributions, this gives universality for $\alpha > 2/17$, and when the third moment vanishes, this threshold improves to $\alpha > 1/10$. These results substantially broaden the known range of universality for directed polymers in the intermediate disorder regime.