The Critical 2d Stochastic Heat Flow and Related Models

2024/12/13 by Francesco Caravenna, Caravenna, Francesco, Rongfeng Sun +3 · 4 citations
Economics, Econometrics and Finance · Engineering · #60H15 #82D60 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary: 82B44 #Probability (math.PR) #Process Optimization and Integration #Reservoir Engineering and Simulation Methods #Secondary: 35R60 #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2412.10311

openalex publication_date 2024/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In these lecture notes, we review recent progress in the study of the stochastic heat equation and its discrete analogue, the directed polymer model, in spatial dimension 2. It was discovered that a phase transition emerges on an intermediate disorder scale, with Edwards-Wilkinson (Gaussian) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and named the critical 2d stochastic heat flow. This gives a meaning to the solution of the stochastic heat equation in the critical dimension 2, which lies beyond existing solution theories for singular SPDEs. We outline the proof ideas, introduce the key ingredients, and discuss related literature on disordered systems and singular SPDEs. A list of open questions is also provided.

Cited by

• Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers
• Conditional GMC within the stochastic heat flow
• Sharp moment and upper tail asymptotics for the critical 2d Stochastic Heat Flow
• An upper bound of the lower tail of the mass of balls under the critical 2d stochastic heat flow

Related

• The Critical 2d Stochastic Heat Flow
• From disordered systems to the Critical 2D Stochastic Heat Flow
• Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers
• Singularity and regularity of the critical 2D Stochastic Heat Flow
• Quasi-critical fluctuations for 2d directed polymers