A central limit theorem for two-dimensional directed polymers with critical spatial correlation

On the 1+2 dimensional lattice, we consider a directed polymer in a random Gaussian environment that is independent in time and correlated in space. The spatial correlation is supposed to decay as $(\log |x|)^a /|x|^{2}$, $a>-1$, where the square in the polynomial is known to be critical (Lacoin, Ann. Prob. (2011)). We introduce an intermediate regime of temperature $\beta_N \propto \hat \beta/(\log N)^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{\beta_N}$ converges in distribution towards a Gaussian random variable if $\hat \beta\in (0,\hat \beta_c)$, whereas $W_N^{\beta_N}$ vanishes for $\hat \beta\geq \hat \beta_c$. The variance of the limiting Gaussian distribution, which is given by an inverse Bessel function, is determined by an induction scheme whose multi-scale dependence reflects the critical nature of the correlation. The Gaussianity of the limit follows from a decoupling argument of Cosco, Donadini (2024+).