On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window

Abstract

The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on R^2, extending previous work by Bertini and Cancrini.

arXiv comment50 pages, 2 figures. Final version to appear in Commun. Math. Phys

References in corpus (6)

• A theory of regularity structures
• Free-energy distribution of the directed polymer at high temperature
• The intermediate disorder regime for directed polymers in dimension $1+1$
• Probability distribution of the free energy of a directed polymer in a random medium
• The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher
• Multiparticle Schrodinger operators with point interactions in the plane

Cited by in corpus (11)

• Moments of the 2D SHE at criticality
• A forward-backward SDE from the 2D nonlinear stochastic heat equation
• The scaling limit of the directed polymer with power-law tail disorder
• The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos
• Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two
• Moments of the 2d directed polymer in the subcritical regime and a generalisation of the Erdös-Taylor theorem
• The random heat equation in dimensions three and higher: the homogenization viewpoint
• Gaussian Limits for Subcritical Chaos
• Mesoscopic averaging of the two-dimensional KPZ equation
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime
• The critical disordered pinning measure