On the Moments of the $$(2+1)$$-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos

doi:10.1007/s00220-019-03527-z

Cited by 42 publications

(51 citation statements)

References 24 publications

(43 reference statements)

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“…It is important to understand whether the condition (1.5) is necessary in the statement of Theorem 1.1. In light of recent results about the (2 + 1)-dimensional stochastic heat equation with multiplicative noise [8,9,16], we believe that there are values of ν, λ and D for which Theorem 1.1 is not valid (specifically, when the effective coupling constant is large). Understanding this is at present out of the reach of the technology developed in this paper.…”

Section: )mentioning

confidence: 90%

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Chatterjee¹,

Dunlap²

2020

Ann. Probab.

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The (d + 1)-dimensional KPZ equation is the canonical model for the growth of rough d-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for d = 1 has been achieved in recent years, and the case d ≥ 3 has also seen some progress. The most physically relevant case of d = 2, however, is not very wellunderstood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the d = 2 case is neither ultraviolet superrenormalizable like the d = 1 case nor infrared superrenormalizable like the d ≥ 3 case. Moreover, unlike in d = 1, the Cole-Hopf transform is not directly usable in d = 2 because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article we show the existence of subsequential scaling limits as ε → 0 of Cole-Hopf solutions of the (2 + 1)dimensional KPZ equation with white noise mollified to spatial scale ε and nonlinearity multiplied by the vanishing factor | log ε| − 1 2 . We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a non-vanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in 2 + 1 dimensions.where ν, λ and D are strictly positive parameters andẆ denotes a standard space-time white noise on the two-dimensional torus T 2 = R 2 /Z 2 . More precisely, we define W to be a cylindrical Wiener process on L 2 (T 2 ) whose covariance operator is the identity, as in [13] or [3], and thenẆ is its (distributional) derivative in time. Thus, formally we have

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Section: )mentioning

confidence: 90%

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Chatterjee¹,

Dunlap²

2020

Ann. Probab.

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“…At the critical value β = √ 2π, the early work of Bertini-Cancrini [1] identified the limiting covariance function of the corresponding stochastic heat equation. While the limiting distribution remains an open question, we refer to the work of [5,16] in this direction.…”

Section: Introductionmentioning

confidence: 99%

Gaussian fluctuations from the 2D KPZ equation

2019

Stoch PDE: Anal Comp

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“…Its limiting correlation structure was first identified in [BC98] through a different regularisation of the 2d SHE (1.6) (mollifying the noise 9 W instead of discretizing space and time). In [CSZ19b], the third moment of the averaged random field U N pt, ϕq :" ş U N pt, xq ϕpxq dx, for test functions ϕ, was computed and shown to converge to a finite limit as N Ñ 8, which implies that all subsequential limits of U N have the same correlation structure identified in [BC98] (tightness is trivial since ErU N s " 1). Subsequently, [GQT21] identified the limit of all moments of U N pt, ϕq (see also the more recent work [Che21]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Critical 2d Stochastic Heat Flow

Caravenna¹,

Sun²,

Zygouras³

2021

Preprint

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We consider directed polymers in random environment in the critical dimension d " 2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R 2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

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On the Moments of the $$(2+1)$$-Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

Cited by 42 publications

References 24 publications

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Gaussian fluctuations from the 2D KPZ equation

The Critical 2d Stochastic Heat Flow

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