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The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off

Georgiana Chatzigeorgiou, Peter Morfe, Felix Otto, Lihan Wang

TL;DR

The paper analyzes the large-time behavior of a 2D passive tracer with a divergence-free, UV-cutoff Gaussian drift expressed as a curl of a Gaussian free field. By introducing an infrared cut-off at scale $L$ and formulating an associated homogenization problem, the authors show that the mean-squared displacement grows like $t\sqrt{\ln t}$ and that the effective diffusivity satisfies $\lambda_L^2 \approx 1+\varepsilon^2\ln L$ up to controllable constants. A central feature is the construction of augmented correctors and stationary proxy fields that propagate across scales via a geometric recursion, enabling sharp energy estimates and a rigorous link between IR-scale diffusivity and real-time diffusion. The results provide a PDE-based, real-time verification of the predicted super-diffusive behavior in 2D and offer a robust, extensible framework for other divergence-free random drifts. Overall, the paper advances rigorous understanding of convection-enhanced diffusion in critical dimensions through stochastic homogenization with infra-red scaling.

Abstract

We analyze the large-time asymptotics of a passive tracer with drift equal to the curl of the Gaussian free field in two dimensions with ultra-violet cut-off at scale unity. We prove that the mean-squared displacement scales like $t \sqrt{\ln t}$, as predicted in the physics literature and recently almost proved by the work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022), which uses mathematical-physics type analysis in Fock space. Our approach involves studying the effective diffusivity $λ_{L}$ of the process with an infra-red cut-off at scale $L$, and is based on techniques from stochastic homogenization.

The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off

TL;DR

The paper analyzes the large-time behavior of a 2D passive tracer with a divergence-free, UV-cutoff Gaussian drift expressed as a curl of a Gaussian free field. By introducing an infrared cut-off at scale and formulating an associated homogenization problem, the authors show that the mean-squared displacement grows like and that the effective diffusivity satisfies up to controllable constants. A central feature is the construction of augmented correctors and stationary proxy fields that propagate across scales via a geometric recursion, enabling sharp energy estimates and a rigorous link between IR-scale diffusivity and real-time diffusion. The results provide a PDE-based, real-time verification of the predicted super-diffusive behavior in 2D and offer a robust, extensible framework for other divergence-free random drifts. Overall, the paper advances rigorous understanding of convection-enhanced diffusion in critical dimensions through stochastic homogenization with infra-red scaling.

Abstract

We analyze the large-time asymptotics of a passive tracer with drift equal to the curl of the Gaussian free field in two dimensions with ultra-violet cut-off at scale unity. We prove that the mean-squared displacement scales like , as predicted in the physics literature and recently almost proved by the work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022), which uses mathematical-physics type analysis in Fock space. Our approach involves studying the effective diffusivity of the process with an infra-red cut-off at scale , and is based on techniques from stochastic homogenization.
Paper Structure (13 sections, 13 theorems, 190 equations)

This paper contains 13 sections, 13 theorems, 190 equations.

Table of Contents

  1. Setting and statement of main result
  2. Related work
  3. Heuristics: an ODE for $\lambda_L$ as function of $L$
  4. Homogenization error estimate based on augmented corrector $(\phi_L,\sigma_L)$
  5. Construction of a stationary proxy $(\tilde{\phi}_L,\tilde{\sigma}_L)$ for $(\phi_L,\sigma_L)$
  6. Estimate of the stationary proxy $(\tilde{\phi}_L,\tilde{\sigma}_L)$
  7. Estimate of the approximation error $f_L$
  8. Proof of Theorem \ref{['t:main']}
  9. Proof of Theorem \ref{['t:laplace']}
  10. Heuristic Derivation of the Scaling Law
  11. Preliminaries from Homogenization Theory
  12. The Effective Diffusivity
  13. Flux Corrector $\sigma_{L}$

Key Result

Theorem 1.1

For every $\epsilon^2 < \infty$, and every unit vector $\xi$, there exists a constant $C_{\epsilon}$ such that for every $t \geq 1$ Moreover, $C_{\epsilon} =1+ O(|\epsilon|)$ as $\epsilon^2\to 0$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 5.1
  • Lemma 5.2
  • ...and 7 more