Sharp moment and upper tail asymptotics for the critical 2d Stochastic Heat Flow

2025/07/29 by Ganguly, Shirshendu, Nam, Kyeongsik · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.22029

Abstract

While 1+1 dimensional growth models in the Kardar-Parisi-Zhang universality class have witnessed an explosion of activity, higher dimensional models remain much less explored. The special case of 2+1 dimensions is particularly interesting as it is, in physics parlance, neither ultraviolet nor infrared super-renormalizable. Canonical examples include the stochastic heat equation (SHE) with multiplicative noise and directed polymers. The models exhibit a weak to strong disorder transition as the inverse temperature, up to a logarithmic (in the system size) scaling, crosses a critical value. While the sub-critical picture has been established in detail, very recently [CSZ '23] constructed a scaling limit of the critical 2+1 dimensional directed polymer partition function, termed as the critical 2d Stochastic Heat Flow (SHF), a random measure on ℝ². The SHF is expected to exhibit a rich intermittent behavior and consequently a rapid growth of its moments. The h^th moment was known to grow at least as exp(Ω(h²)) (a consequence of the Gaussian correlation inequality) and at most as exp(exp (O(h²))). The true growth rate, however, was predicted to be exp(exp (Θ(h))) in the late nineties [R '99]. In this paper we prove a lower bound of the h^th moment which matches the predicted value, thereby exponentially improving the previous lower bound. We also obtain rather sharp bounds on its upper tail. The key ingredient in the proof involves establishing a new connection of the SHF and moments thereof to the Gaussian Free Field (GFF) on related Feynman diagrams. This connection opens the door to the rich algebraic structure of the GFF to study the SHF. Along the way we also prove a new monotonicity property of the correlation kernel for the SHF as a consequence of the domain Markov property of the GFF.

Citations

• Conditional GMC within the stochastic heat flow
• Stochastic motions of the two-dimensional many-body delta-Bose gas, II: Many-δ motions
• Singularity and regularity of the critical 2D Stochastic Heat Flow
• The Critical 2d Stochastic Heat Flow and Related Models
• On the moments of the mass of shrinking balls under the Critical 2d Stochastic Heat Flow
• Stochastic heat flow by moments
• Continuum polymer measures corresponding to the critical 2d stochastic heat flow
• The tail distribution of the partition function for directed polymer in the weak disorder phase
• Directed polymers in a random environment: a review of the phase transitions
• Fractal geometry of the PAM in 2D and 3D with white noise potential
• A multivariate extension of the Erdös-Taylor theorem
• Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- I
• Random metric geometries on the plane and Kardar-Parisi-Zhang universality
• The Critical 2d Stochastic Heat Flow
• Fractal Geometry of the Valleys of the Parabolic Anderson Equation
• Law of Iterated Logarithms and Fractal Properties of the KPZ Equation
• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Gaussian fluctuations from the 2D KPZ equation
• The two-dimensional KPZ equation in the entire subcritical regime
• Constructing a solution of the (2+1)-dimensional KPZ equation
• The Dickman subordinator, renewal theorems, and disordered systems
• The scaling limit of the KPZ equation in space dimension 3 and higher
• Universality in marginally relevant disordered systems
• A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions
• Gaussian multiplicative chaos and applications: a review
• Intermittence and nonlinear parabolic stochastic partial differential equations
• A Condensation of Interacting Bosons in Two Dimensional Space

Cited by

• From disordered systems to the Critical 2D Stochastic Heat Flow
• Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers
• 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
• The Critical 2d Stochastic Heat Flow and Related Models
• Stochastic heat flow by moments
• From disordered systems to the Critical 2D Stochastic Heat Flow
• Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers