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Long range voter models and dynamical fractional Brownian motion

Reuben Drogin

TL;DR

This work analyzes a voter model on $\mathbb{Z}$ with long-range interactions drawn from a measure $\mu$ in the domain of attraction of an $\alpha$-stable law. Under a spacetime rescaling, the authors prove convergence to a one-parameter family of fractional Gaussian fields $W_\alpha$, i.e., a dynamical fractional Brownian motion in space with fractional Gaussian noise in time; precisely, along a time slice the field converges to a fractional Brownian motion with Hurst index $1/2+\alpha/2$. The key techniques combine a Lindeberg swapping argument with novel heat-kernel and moment bounds for the associated random graph $\mathcal{G}_\mu$, enabling Gaussianity despite long-range dependence. The results extend classical short-range scaling (Gaussian free field limits) to a long-range setting and provide a framework for rates of convergence and potential higher-dimensional generalizations, with implications for invariant measures $\nu_p$ of the voter model and their scaling limits.

Abstract

We study the voter model on Z with long-range interactions, as proposed by Hammond and Sheffield. We show a spacetime rescaling converges to a fractional Gaussian free field, which can be viewed as a one-parameter family of fractional Brownian motions. As a consequence, we obtain that long-range voter models rescale to fractional Gaussian noise. The argument uses the Lindeberg swapping technique and heat kernel estimates for random walks with jump distributions in the domain of attraction of a stable law.

Long range voter models and dynamical fractional Brownian motion

TL;DR

This work analyzes a voter model on with long-range interactions drawn from a measure in the domain of attraction of an -stable law. Under a spacetime rescaling, the authors prove convergence to a one-parameter family of fractional Gaussian fields , i.e., a dynamical fractional Brownian motion in space with fractional Gaussian noise in time; precisely, along a time slice the field converges to a fractional Brownian motion with Hurst index . The key techniques combine a Lindeberg swapping argument with novel heat-kernel and moment bounds for the associated random graph , enabling Gaussianity despite long-range dependence. The results extend classical short-range scaling (Gaussian free field limits) to a long-range setting and provide a framework for rates of convergence and potential higher-dimensional generalizations, with implications for invariant measures of the voter model and their scaling limits.

Abstract

We study the voter model on Z with long-range interactions, as proposed by Hammond and Sheffield. We show a spacetime rescaling converges to a fractional Gaussian free field, which can be viewed as a one-parameter family of fractional Brownian motions. As a consequence, we obtain that long-range voter models rescale to fractional Gaussian noise. The argument uses the Lindeberg swapping technique and heat kernel estimates for random walks with jump distributions in the domain of attraction of a stable law.
Paper Structure (15 sections, 13 theorems, 112 equations, 3 figures)

This paper contains 15 sections, 13 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary and Background
  3. Statement of Results
  4. Discussion of the Scaling Limit $W_{\alpha}$
  5. A Brief Proof Sketch
  6. Notation
  7. Acknowledgements
  8. Number of Components of $\mathcal{G}_\mu$
  9. Asymptotics for the Variance
  10. Estimates on the Random Graph $\mathcal{G}_\mu$
  11. A Heat Kernel Estimate
  12. Moment Bounds on Connected Components
  13. Gaussianity via Lindeberg Swapping
  14. Proof of Theorem \ref{['thm:fbm']}
  15. Applying Theorem \ref{['thm: main']} to the Voter Model

Key Result

Theorem 1.2

Let $p,\alpha\in(0,1)$, $\mu\in \Gamma_\alpha$, and $\xi:\mathbb{Z}\to\{+1,-1\}$ have law $\nu_p$. Define $F^n_p\in (\mathcal{S}(\mathbb{R}))^{'}$ to be the random tempered distribution defined by for any $\phi\in \mathcal{S}(\mathbb{R})$, where $\sigma_n$ is the deterministic sequence given in Theorem thm: main. Then as $n\to\infty$. In the language of generalized random fields, $F_p^n$ converg

Figures (3)

  • Figure 1: Space-time plots of long-range and short-range voter models in one dimension.
  • Figure 2: A connected component in a sample of $\mathcal{G}_\mu$, for a truncated $\mu\in\Gamma_{1/2}$
  • Figure 3: In (A), the left the ancestral lines coalesce one at a time. In (B) the ancestral lines coalesce pairwise.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 24 more