Free energy and phase transition for 2D directed polymers with critical spatial correlations

We study the two-dimensional directed polymer model in a Gaussian environment which is independent in time and spatially correlated, with covariances $h(x)$ either summable or with a critical decay, satisfying $h(x)\sim (\log \mathrm{\mid }x\mathrm{\mid }{)}^a\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ for some $a>-1$. We determine the precise high-temperature asymptotics of the free energy, confirming a conjecture of Lacoin (Ann. Probab. 2011), later refined by Cosco, Cottini and Donadini (2025). We also establish a phase transition for the diffusively rescaled partition functions: below some critical point they converge to the Lebesgue measure, while above it they converge to zero. A key feature of our approach is that both results are obtained using only second-moment estimates and are based on a simplified change-of-measure argument in the supercritical regime, that may prove useful for other disordered models.