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Quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster for $d \ge 1$

Stein Andreas Bethuelsen, Matthias Birkner, Andrej Depperschmidt, Timo Schlüter

TL;DR

This work extends the quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster from higher dimensions to all dimensions $d\ge1$, building on and unifying prior annealed and quenched results. The authors develop a dimension-free framework leveraging environment exposure, regeneration times, and a refined quenched-annealed comparison to control discrepancies between quenched and annealed laws, ultimately proving the existence of a unique invariant measure $Q$ absolutely continuous with respect to the environment and a quenched local limit theorem characterized by a density factor $\phi=dQ/d\mathbb{P}$ via $P_\omega^{(o,0)}(X_n=x) \approx \mathbb{P}^{(o,0)}(X_n=x)\phi(\sigma_{(x,n)}\omega)$ as $n\to\infty$. The approach, combining environment seen-from-the-particle techniques with refined probabilistic estimates for box events and walk intersections, yields robust control over low-dimensional cases and provides a versatile toolkit for random walks in dynamic random environments. The results have implications for understanding ballistic processes and ancestral lineages in fluctuating populations, and they broaden the applicability of quenched LLTs in random media. Key methodological contributions include an extended quenched-annealed comparison (via Theorem 3.24 Ersatz), the construction of an invariant environment measure with concentration, and a rigorous local limit statement that remains valid in $d=1$ and $d=2$.

Abstract

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions $d+1$ with $d\geq 1$ being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all $d\geq 1$ and a quenched local limit theorem under the assumption $d\geq 3$. Here we show that the latter result also holds for all $d \ge 1$.

Quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster for $d \ge 1$

TL;DR

This work extends the quenched local limit theorem for a directed random walk on the backbone of a supercritical oriented percolation cluster from higher dimensions to all dimensions , building on and unifying prior annealed and quenched results. The authors develop a dimension-free framework leveraging environment exposure, regeneration times, and a refined quenched-annealed comparison to control discrepancies between quenched and annealed laws, ultimately proving the existence of a unique invariant measure absolutely continuous with respect to the environment and a quenched local limit theorem characterized by a density factor via as . The approach, combining environment seen-from-the-particle techniques with refined probabilistic estimates for box events and walk intersections, yields robust control over low-dimensional cases and provides a versatile toolkit for random walks in dynamic random environments. The results have implications for understanding ballistic processes and ancestral lineages in fluctuating populations, and they broaden the applicability of quenched LLTs in random media. Key methodological contributions include an extended quenched-annealed comparison (via Theorem 3.24 Ersatz), the construction of an invariant environment measure with concentration, and a rigorous local limit statement that remains valid in and .

Abstract

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions with being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all and a quenched local limit theorem under the assumption . Here we show that the latter result also holds for all .

Paper Structure

This paper contains 16 sections, 20 theorems, 143 equations.

Table of Contents

  1. Introduction and outline of the paper
  2. The model and main results
  3. A first quenched-annealed comparison result
  4. Preliminary annealed estimates
  5. Bounds on the intersection of two random walks
  6. Proof of Lemma \ref{['lem:Prop 3.11 Ersatz']}
  7. Proof of Theorem \ref{['thm: Theorem 3.24 Ersatz']}
  8. Proofs of the main results
  9. An improved quenched annealed-comparison result
  10. Existence of a measure absolutely continuous with respect to the environment: proof of Theorem \ref{['thm:main1']}
  11. The local limit theorem: proof of Theorem \ref{['thm:main2']}
  12. Proofs of preliminary annealed estimates
  13. Proof of Lemma \ref{['lem:annealed_derivative_estimates']}
  14. Proof of Lemma \ref{['lem:additional_annealed_estimate']}
  15. Proof of Lemma \ref{['lem:lem3.6SS']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $d\ge 1$ and $p \in (p_c,1]$. Then there exists a unique measure $Q$ on $\Omega$ which is invariant with respect to the point of view of the particle satisfying $Q \ll \mathbb{P}$ and the concentration property eq:concQ:

Theorems & Definitions (44)

  • Theorem 2.1: Extension of BethuelsenBirknerDepperschmidtSchluter
  • Theorem 2.2: Quenched local limit theorem, extension of BethuelsenBirknerDepperschmidtSchluter
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Theorem 3.2: Extension of SteibersPhD2017
  • Lemma 3.3: Extension of Proposition 3.11 in SteibersPhD2017
  • Remark 3.4
  • Lemma 3.5: Extension of Lemma 3.1 in BethuelsenBirknerDepperschmidtSchluter
  • Lemma 3.6: Extension of Lemma 3.2 in BethuelsenBirknerDepperschmidtSchluter
  • ...and 34 more