Edwards-Wilkinson fluctuations for the directed polymer in the full $L^2$-regime for dimensions $d\ge 3$

Abstract

We prove that in the full $L^2$-regime the partition function of the directed polymer model in dimensions $d\ge 3$, if centered, scaled and averaged with respect to a test function $\phi \in C_c({\mathbb{R}}^d)$, converges in distribution to a Gaussian random variable with explicit variance. Introducing a new idea in this context of a martingale difference representation, we also prove that the log-partition function, which can be viewed as a discretisation of the KPZ equation, exhibits the same fluctuations, when centered and averaged with respect to a test function. Thus, the two models fall within the Edwards-Wilkinson universality class in the full $L^2$-regime, a result that was only established, so far, for a strict subset of this regime in $d\ge 3$.

arXiv comment46 pages, 2 figures, final version to appear at Ann. Inst. Henri Poincare

References in corpus (8)

• Central limit theorems for sequences of multiple stochastic integrals
• Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
• A Local limit theorem for directed polymers in random media: the continuous and the discrete case
• Universality for the pinning model in the weak coupling regime
• Space-time fluctuation of the Kardar-Parisi-Zhang equation in $d\ge 3$ and the Gaussian free field
• Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder
• Gaussian fluctuations for the directed polymer partition function for $d\ge 3$ and in the whole $L^2$-region
• Scaling limits of disordered systems and disorder relevance

Cited by in corpus (4)

• Weak coupling limit of the Anisotropic KPZ equation
• Universality of deterministic KPZ
• Local KPZ behavior under arbitrary scaling limits
• Central limit theorems for the (2+1)-dimensional directed polymer in the weak disorder limit