A central limit theorem for two-dimensional directed polymers with critical spatial correlation
Clément Cosco, Francesca Cottini, Anna Donadini
TL;DR
This work analyzes a 1+2 dimensional directed polymer in a Gaussian environment with time-independent but spatially critical correlations h(x) ~ (\\log |x|)^a / |x|^2, a>-1. By tuning the inverse temperature β_N at the scale β_N ∝ (\\log N)^{-(a+2)/2} into an intermediate disorder regime, the authors establish a phase transition: for hat{β} below a critical z_a the log-partition log W_N^{β_N} converges to a Gaussian with variance λ^2 determined by a Bessel-function-based expression, while at or above the threshold the partition function vanishes; the limiting Gaussian variance is computed via a multi-scale induction that reflects the critical spatial correlation. A central technical achievement is the sharp second-moment analysis, including a generalized Erdős–Taylor result for the sum over h along the random walk when correlations are critical, and a robust log-normal CLT for the partition function built upon an L^2 decoupling across time blocks. The results illuminate the precise fluctuation structure in the subcritical regime and open avenues for critical-limit SPDE-type descriptions under colored noise, with potential stochastic heat-flow analogues at the critical point.
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 $β_N \propto \hat β/(\log N)^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{β_N}$ converges in distribution towards a Gaussian random variable if $\hat β\in (0,\hat β_c)$, whereas $W_N^{β_N}$ vanishes for $\hat β\geq \hat β_c$. 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+).