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

Abstract

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 ${\hat{I}}_N^2\propto {\hat{\hat{I}}}^2/(\log N{)}^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{{\hat{I}}_N^2}$ converges in distribution towards a Gaussian random variable if ${\hat{\hat{I}}}^2\in (0,{\hat{\hat{I}}}_c^2)$, whereas $W_N^{{\hat{I}}_N^2}$ vanishes for ${\hat{\hat{I}}}^2\ge {\hat{\hat{I}}}_c^2$. 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+).

26 pages