Scaling limit of 1+1 dimensional directed polymer with power-law tail and spatial correlated noise

Abstract

We study a $(1+1)$-dimensional directed polymer in a spatially correlated random environment generated by power-law tail variables: $\omega (i,x)={\sum }_{y\in \mathbb{Z}}{\psi }_{y-x}\xi (i,y),{\psi }_y\sim {\lambda }_r|y{|}^{-r},r\in (1/2,1)$, where the variables $\xi (i,y)$ are i.i.d. and have a regularly varying right tail with exponent $\alpha >2$. The spatial covariance of the environment has long-range decay with Hurst parameter $H=\frac{3}{2}-r\in (1/2,1)$. We identify the limiting fluctuations of the log-partition function in the intermediate disorder regime and show that the critical tail exponent is ${\alpha }_c=\frac{3}{H}=\frac{6}{3-2r}$. When $\alpha >{\alpha }_c$, the model has the same scaling limits as the corresponding Gaussian spatially correlated polymer: if ${\beta }_N{N}^{H/2}\to \beta \in (0,\infty )$, the centered log-partition function converges to the logarithm of the solution of the stochastic heat equation driven by fractional spatial noise; if ${\beta }_N{N}^{H/2}\to 0$, its normalized fluctuation converges to a centered Gaussian law. In the regime $2<\alpha \le {\alpha }_c$, at the scale ${\beta }_N{N}^{H/2}=\beta N^{H/2}/l(N^{3/2})$, the log-partition function still satisfies Gaussian fluctuation. The main ingredient is a truncation comparison argument adapted to long-range moving-average environments, together with an invariance principle for polynomial chaos. Due to the non-locality of the environments, we perform a far-near field analysis, as well as multiscale analysis, to prove that the truncated version does not change the log-partition function at the corresponding scales.

48pages