Intermediate disorder for directed polymers with space-time correlations
Shalin Parekh
TL;DR
The paper analyzes polymer models in strongly mixing space–time environments under intermediate disorder and proves convergence of the rescaled partition function to the Itô solution of the stochastic heat equation with multiplicative noise and a shear term: ∂_t U(t,x)= ½∂_x^2U + v∂_xU + σUξ, where ξ is space–time white noise. The authors develop a moment-based approach using cumulant expansions of the stationary sequence η_t = ω_{−t,R(t)} to obtain the renormalization constants c_N and parameters v and σ without assuming Gaussianity. They decompose higher moments into diagonal and off-diagonal contributions, identify Brownian and local-time limiting objects, and establish exponential moment bounds to control all processes. A recent flow-based classification result is leveraged to upgrade moment convergence to the Itô-Walsh SHE limit, providing integrated and endpoint convergence statements for lattice, semi-discrete, and continuous polymer models under space–time correlations. This yields a robust framework for intermediate-disorder limits beyond Gaussian or white-in-time environments, linking polymer partition functions to linear SHE with a shear drift.
Abstract
We revisit a result of Hairer-Shen on polymer-type approximations for the stochastic heat equation with a multiplicative noise (SHE) in $d=1$. We consider a general class of polymer models with strongly mixing environment in space and time, and we prove convergence to the Itô solution of the SHE (modulo shear). The environment is not assumed to be Gaussian, nor is it assumed to be white-in-time. Instead of using regularity structures or paracontrolled products, we rely on simpler moment-based characterizations of the SHE to prove the convergence. However, the price to pay is that our topology of convergence is weak.