The heat and the landscape I

Abstract

Heat flows in 1+1 dimensional stochastic environment converge after scaling to the random geometry described by the directed landscape. In this first part, we show that the O'Connell-Yor polymer and the KPZ equation converge to the KPZ fixed point. The key is that one-dimensional Baik-Ben Arous-Peche statistics characterize the KPZ fixed point. This yields a general and elementary method that shows convergence based on previously established limit theorems. Independently, at the same time and place Quastel and Sharkar gave an unrelated proof of KPZ convergence. The methods invite extensions in different directions: ours to polymer models, and theirs to interacting particle systems.

arXiv comment20 pages

References in corpus (6)

• The KPZ fixed point
• Limits of spiked random matrices I
• The directed landscape
• Littelmann paths and brownian paths
• Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment
• Bulk properties of the Airy line ensemble

Cited by in corpus (7)

• Fluctuations of the log-gamma polymer free energy with general parameters and slopes
• Weak coupling limit of the Anisotropic KPZ equation
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• The geometric Burge correspondence and the partition function of polymer replicas
• Brownian regularity for the KPZ line ensemble
• KPZ on torus: Gaussian fluctuations
• Tightness of $(H,H^{RW})$-Gibbsian line ensembles