The Critical 2d Stochastic Heat Flow and Related Models

TL;DR

The work analyzes the critical two-dimensional stochastic heat flow (SHF) arising from the stochastic heat equation (SHE) and its directed polymer discretisation in the intermediate disorder regime. It establishes a universal, non-Gaussian limit SHF(\vartheta) at criticality by combining polynomial chaos analysis, a coarse-grained Lindeberg principle, and a renewal/Dickman-subordinator framework to control second and higher moments. The results include precise first and second moment formulas, a sharp covariance kernel, and higher-moment bounds, together with shift/scale covariance, flow properties, and a rigorous argument showing SHF is not Gaussian Multiplicative Chaos. These findings illuminate the critical behavior of disordered systems in dimension two and provide a robust continuum limit for singular SPDEs at the marginal boundary, with potential implications for broader KPZ-like universality and stochastic interface models.

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.

Figures (5)

• Figure 4: The pictures above illustrate the critical $2d$ stochastic heat flow, $\mathsf{SHF}(\vartheta)$, and its scaling properties. The top-right picture and the two bottom pictures are zoomed-in snapshots of the upper-left picture, which is a simulation of $\mathsf{SHF}(\vartheta)$ with $\vartheta=0$. The singularity of the critical $2d$ SHF w.r.t. the Lebesque measure can be seen from the upper-left picture. The zoomed-in snapshots demonstrate the scaling covariance of $\mathsf{SHF}(\vartheta)$ stated in \ref{['eq:scaling']}. In particular, the zoomed-in fields appear to be smoother than the original field, since \ref{['eq:scaling']} shows that zooming in has the effect of decreasing the disorder strength $\vartheta$.
• Figure 5: Space-time is partitioned into mesoscopic boxes ${\mathcal{B}} _{\varepsilon N}(\mathsf i,\mathsf a)$ as in \ref{['Bbox']}. Given $\Gamma=((n_1,z_1),...,(n_r, z_r))$, the sequence of space-time points associated with a given term in the chaos expansion in \ref{['eq:Zppchaos']}, the boxes containing the dotted lines are the ones that intersect $\Gamma$. The starting point $(d, x)\in \Gamma$ and the end point $(f, y)\in \Gamma$ of each dotted line are the points of entry and exit in $\Gamma$ for that mesoscopic time interval ${\mathcal{T}} _{\varepsilon N}(\mathsf i)$. Each dotted line contributes a factor $X_{d, f}(x, y)$ to the corresponding term in the chaos expansion, while each solid line contributes a random walk transition kernel.
• Figure 6: A simplified coarse-graining that defines ${\mathcal{Z}} _{N,\varepsilon}^{({\rm mock}-\mathrm{cg})}(\varphi, \psi | \Theta_{N, \varepsilon})$. A coarse-grained disorder variable $\Theta_{N, \varepsilon}(\mathsf i, \vec{\mathsf a})$ is defined for each visited mesoscopic time interval ${\mathcal{T}} _{\varepsilon N}(\mathsf i)$. The random walk transition kernel connecting ${\mathcal{T}} _{\varepsilon N}(\mathsf i)$ and ${\mathcal{T}} _{\varepsilon N}(\mathsf j)$, with $\mathsf i<\mathsf j$, is replaced by a heat kernel connecting the corner of the mesoscopic spatial box of exit from ${\mathcal{T}} _{\varepsilon N}(\mathsf i)$ to the corner of the box of entry in ${\mathcal{T}} _{\varepsilon N}(\mathsf j)$, represented here by a solid line. Such replacements can lead to poor approximations if $\mathsf j-\mathsf i<K_\varepsilon$.
• Figure 7: A more accurate coarse-graining approximation. Since the second and third visited boxes are separated by less than $K_\varepsilon$ mesoscopic time intervals, they need to be grouped together as part of a single coarse-grained disorder variable $\Theta_{N, \varepsilon}(\vec{\mathsf i}, \vec{\mathsf a})$.
• Figure : A picture of the critical $2d$ stochastic heat flow is shown on the left, and on the right, a resembling natural landscape located in central Greece and named Meteora (https://en.wikipedia.org/wiki/Meteora#/media/File:Meteora's_monastery_2.jpg by https://www.flickr.com/photos/antistath/, https://creativecommons.org/licenses/by-sa/4.0/). The picture of the critical $2d$ stochastic heat flow has been obtained by simulating the partition function of the directed polymer model. The plateaus in the simulation are a result of a truncation of high peaks.

Theorems & Definitions (33)

• Lemma 2.1: Transition in the second moment
• Corollary 2.2: Exponential time scale
• Theorem 3.1: Limit of individual partition function
• Remark 3.2
• Theorem 3.3: Edwards-Wilkinson fluctuation for DPM
• Theorem 4.1: Edwards-Wilkinson fluctuation for subcritical $2d$ KPZ - CSZ20G20
• Theorem 4.2
• Remark 4.3
• Theorem 5.1: 1-point statistics for the mollified SHE
• Theorem 5.2: Edwards-Wilkinson fluctuation for the mollified SHE