Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A case study of the long-time behavior of the Gaussian local-field equation

Kevin Hu, Kavita Ramanan

TL;DR

The paper studies the Gaussian local-field setting for κ-regular graphs, focusing on the long-time behavior and the link between non-Markovian LFE dynamics and the Markov MLFE. By specializing to quadratic potentials, it proves well-posedness of the κ-GMLFE and derives explicit Gaussian solutions for the 2-GLFE, proving exponential convergence to a stationary Gaussian π. It then shows that both the 2-GLFE and the 2-GMLFE converge to π at exponential rates and that the two approaches are exponentially close, providing evidence for the conjectured LFE–MLFE correspondence in this Gaussian affine-drift regime. The work further establishes commuting limits for the n-cycle and develops a nonlinear Riccati framework to guarantee uniqueness, culminating in a robust entropic/H-theorem-based analysis for long-time behavior. These results contribute a rigorous bridge between non-Markovian LFE dynamics and tractable Markov approximations, with explicit rates and stationary structures.

Abstract

For any integer $κ\geq 2$, the $κ$-local-field equation ($κ$-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on $κ$-regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) $κ$-LFE coincides with that of a certain more tractable Markovian analog, the Markov $κ$-local-field equation. In the present article, we prove this conjecture for the case when $κ= 2$ and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the $n$-cycle (or 2-regular random graph on $n$ vertices), the limits $n \rightarrow \infty$ and $t\rightarrow \infty$ commute. Along the way, we also establish well-posedness of the Markov $κ$-local field equations with affine drifts for all $κ\geq 2$, which may be of independent interest.

A case study of the long-time behavior of the Gaussian local-field equation

TL;DR

The paper studies the Gaussian local-field setting for κ-regular graphs, focusing on the long-time behavior and the link between non-Markovian LFE dynamics and the Markov MLFE. By specializing to quadratic potentials, it proves well-posedness of the κ-GMLFE and derives explicit Gaussian solutions for the 2-GLFE, proving exponential convergence to a stationary Gaussian π. It then shows that both the 2-GLFE and the 2-GMLFE converge to π at exponential rates and that the two approaches are exponentially close, providing evidence for the conjectured LFE–MLFE correspondence in this Gaussian affine-drift regime. The work further establishes commuting limits for the n-cycle and develops a nonlinear Riccati framework to guarantee uniqueness, culminating in a robust entropic/H-theorem-based analysis for long-time behavior. These results contribute a rigorous bridge between non-Markovian LFE dynamics and tractable Markov approximations, with explicit rates and stationary structures.

Abstract

For any integer , the -local-field equation (-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on -regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) -LFE coincides with that of a certain more tractable Markovian analog, the Markov -local-field equation. In the present article, we prove this conjecture for the case when and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the -cycle (or 2-regular random graph on vertices), the limits and commute. Along the way, we also establish well-posedness of the Markov -local field equations with affine drifts for all , which may be of independent interest.

Paper Structure

This paper contains 33 sections, 21 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Gaussian setting and main contributions
  4. Generalizations and Future Directions
  5. Notation
  6. Matrices
  7. Probability measures and equipped functionals
  8. Function spaces
  9. Local-Field Equations
  10. Preliminaries about probability measures on $\kappa$-stars
  11. Markov local-field equation on the $\kappa$-star
  12. Definition of the Markov local-field equation
  13. Well-posedness of the $\kappa$-GMLFE
  14. Gaussian local-field equation on the line.
  15. Definition of the Gaussian local-field equation
  16. ...and 18 more sections

Key Result

Lemma 2.5

Let $\kappa \in \mathbb{N}$ satisfy $\kappa \geq 2$. If $\lambda \in {\mathcal{Q}}_\kappa$ is a centered Gaussian measure, then its covariance matrix $\Sigma_\lambda$ lies in $\mathscr{M}_\kappa$. Conversely, if $\Sigma = \mathsf{M}_\kappa(a, b, c)$ is positive semi-definite for $(a, b, c) \in \math

Figures (1)

  • Figure 1.1: Interchange of $n \rightarrow \infty$ and $t \rightarrow \infty$ limits for the interacting diffusion $Z^n$ in \ref{['e:zOU']}. The dashed convergence follows from classical results for Langevin diffusions described above, and the squiggly convergence follows from Theorem 3.19 of lacker2023marginal. The solid convergence follow from the results of this paper; the right solid arrow is a consequence of Theorem \ref{['t:r:lfeConv']}, and the downward solid arrow follows from Remark \ref{['r:pi_n_to_pi']}.

Theorems & Definitions (60)

  • Conjecture 1
  • Conjecture 2
  • Definition 2.1: Edge marginal
  • Definition 2.2: Symmetric probability measures
  • Remark 2.3: Notation for conditional distributions
  • Definition 2.4: Covariance matrix parametrization
  • Lemma 2.5: Correspondence between measures and covariance matrices
  • Definition 2.6: $\kappa$-regular Markov local-field equation
  • Definition 2.7: Linear growth solution to $\kappa$-MLFE
  • Definition 2.8: Gaussian Markov local-field equation
  • ...and 50 more