Continuum polymer measures corresponding to the critical 2d stochastic heat flow

TL;DR

This paper constructs continuum polymer measures linked to the critical 2d stochastic heat flow, establishing foundational properties and second moment behaviors, revealing Gaussian multiplicative chaos relationships.

Contribution

It introduces a construction of continuum polymer measures for the critical 2d stochastic heat flow and analyzes their second moments and properties.

Findings

Second moments align with Gaussian multiplicative chaos distributions.

Chapman-Kolmogorov relation established for 2d SHF.

Continuum polymer measures exhibit disorder-dependent behavior.

Abstract

We construct continuum directed polymer measures corresponding to the critical 2d stochastic heat flow (2d SHF) introduced by Caravenna, Sun, and Zygouras in their recent article [Inventiones mathematicae 233, 325--460 (2023)]. For this purpose, we prove a Chapman-Kolmogorov relation for the 2d SHF along with a related elementary conditional expectation formula. We explore some basic properties of the continuum polymer measures, with our main focus being on their second moments. In particular, we show that the form of their second moments is consistent with the family of continuum polymer measures, indexed by a disorder strength parameter, having a conditional Gaussian multiplicative chaos distributional interrelationship similar to that previously found in an analogous hierarchical toy model.