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From disordered systems to the Critical 2D Stochastic Heat Flow

Francesco Caravenna, Rongfeng Sun, Nikos Zygouras

Abstract

We review our joint work on the scaling limits of disordered systems, linking the notion of disorder relevance/irrelevance to that of sub/super-criticality of singular SPDEs. This line of research culminated in the construction of the Critical 2D Stochastic Heat Flow (SHF), a universal process which provides a non-trivial solution to the Stochastic Heat Equation in dimension 2, a critical singular SPDE that lies beyond the reach of existing solution theories. The SHF also offers a rare example of a non-Gaussian scaling limit for a disordered system at its phase transition point in the critical dimension.

From disordered systems to the Critical 2D Stochastic Heat Flow

Abstract

We review our joint work on the scaling limits of disordered systems, linking the notion of disorder relevance/irrelevance to that of sub/super-criticality of singular SPDEs. This line of research culminated in the construction of the Critical 2D Stochastic Heat Flow (SHF), a universal process which provides a non-trivial solution to the Stochastic Heat Equation in dimension 2, a critical singular SPDE that lies beyond the reach of existing solution theories. The SHF also offers a rare example of a non-Gaussian scaling limit for a disordered system at its phase transition point in the critical dimension.

Paper Structure

This paper contains 23 sections, 16 theorems, 54 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Disorder relevance/irrelevance, scaling limits, directed polymer, and SHE.
  3. Disordered systems and disorder relevance/irrelevance.
  4. Scaling limits of disorder relevant models.
  5. Directed Polymer Model and Stochastic Heat Equation.
  6. Scaling limits of disorder relevant systems.
  7. Phase transition in the 2D DPM.
  8. 2D SHE and KPZ in the subcritical regime.
  9. Gaussian limit for the subcritical 2D SHE
  10. Gaussian limit for the subcritical 2D KPZ
  11. The Critical 2D Stochastic Heat Flow.
  12. Construction of the SHF.
  13. More historical background
  14. Proof outline of Theorem \ref{['th:main0']}
  15. Properties of the Critical 2D Stochastic Heat Flow.
  16. ...and 8 more sections

Key Result

Theorem 3.2

Let $Z_{\Omega_{\delta} ; \beta_\delta, h_\delta}^{\omega}$ be the disordered partition function defined as in eq:Zom. Suppose the reference measure $\mathbf P_{\Omega_{\delta}}^{\text{ref }}$ satisfies Assumption ass:main for some exponent $\gamma<d/2$. Then choosing we have where $W$ is a white noise on $\Omega$, and $\mathscr{Z}^W_{\Omega; \hat{\beta}, \hat{h}}$ admits the following Wiener-I

Figures (3)

  • Figure 6.1:
  • Figure 7.1: The top-left picture is a simulation of the Critical 2d SHF with $\vartheta=0$. The other three pictures show how this looks when successively zoom into its scales, which effectively corresponds to lowering the disorder parameter $\vartheta$, see the scaling covariance property \ref{['eq:scaling']}.
  • Figure 7.2: The black contours show the level lines of the Critical 2D SHF corresponding to much lower values compared to the peaks.

Theorems & Definitions (27)

  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 4.1: CSZ17b
  • Theorem 4.2: CSZ17b
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Theorem 5.1: Edwards-Wilkinson fluctuations CSZ17b
  • ...and 17 more