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Asymptotics for the percolation threshold of finitary random interlacements in four and higher dimensions

Yijie Bi, Zhenhao Cai, Xinyi Li, Balázs Ráth, Yuan Zhang

TL;DR

This work determines sharp asymptotics for the percolation threshold of finitary random interlacements in dimensions $d\ge 4$ for general length distributions. The authors show that near-critical FRI behavior is well-approximated by a Galton–Watson process whose mean offspring equals the capacity of a random walk killed at the sampled length, yielding a universal limit formula $\lim_{n\to\infty} u_*(\rho_n)\cdot\frac{\mu_2(\rho_n)}{\mu_1(\rho_n)(1+\log\mu_1(\rho_n)\mathbbm{1}_{d=4})}}\varepsilon_d = 1$, with a logarithmic correction in $d=4$. The analysis combines layer-by-layer exploration, truncation of trajectories, coarse-graining into boxes, seeds, and a comparison to finitely dependent percolation via Liggett–Schonmann–Stacey domination. The results extend known asymptotics for geometric lengths and finite-range models to general length distributions, clarifying how long-range dependencies shape the percolation threshold in high dimensions. This provides a precise, robust description of phase transitions in FRI and informs related percolation models with long-range interactions.

Abstract

We establish sharp asymptotic bounds for the critical intensity of the Finitary Random Interlacements (FRI) model in four and higher dimensions with general trajectory length distributions. Our proof reveals that the construction of near-critical FRI clusters in four and higher dimensions is essentially analogous to a Galton-Watson process, whose expected number of offspring corresponds to the capacity of a random walk killed at the given length.

Asymptotics for the percolation threshold of finitary random interlacements in four and higher dimensions

TL;DR

This work determines sharp asymptotics for the percolation threshold of finitary random interlacements in dimensions for general length distributions. The authors show that near-critical FRI behavior is well-approximated by a Galton–Watson process whose mean offspring equals the capacity of a random walk killed at the sampled length, yielding a universal limit formula , with a logarithmic correction in . The analysis combines layer-by-layer exploration, truncation of trajectories, coarse-graining into boxes, seeds, and a comparison to finitely dependent percolation via Liggett–Schonmann–Stacey domination. The results extend known asymptotics for geometric lengths and finite-range models to general length distributions, clarifying how long-range dependencies shape the percolation threshold in high dimensions. This provides a precise, robust description of phase transitions in FRI and informs related percolation models with long-range interactions.

Abstract

We establish sharp asymptotic bounds for the critical intensity of the Finitary Random Interlacements (FRI) model in four and higher dimensions with general trajectory length distributions. Our proof reveals that the construction of near-critical FRI clusters in four and higher dimensions is essentially analogous to a Galton-Watson process, whose expected number of offspring corresponds to the capacity of a random walk killed at the given length.
Paper Structure (10 sections, 27 theorems, 304 equations, 1 figure)

This paper contains 10 sections, 27 theorems, 304 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Proof of the lower bound of (\ref{['new1.4']})
  4. Proof of the upper bound of (\ref{['new1.4']})
  5. Preparation for the Algorithm
  6. The Algorithm
  7. Proof of Proposition \ref{['uniform_low']}.
  8. On the goodness of exploration
  9. Finding seeds for neighboring vertices
  10. Proofs of technical lemmas

Key Result

Theorem 1.4

For any $d\ge4$ and any appropriate family of distributions $(\rho_n)_{n\in\mathbb{N}}$, Here, $\varepsilon_d$ is the coefficient in the asymptotic capacity of random walks, i.e., where ${\rm cap}(A)$ for $A\subset\mathbb Z^d$ represents the capacity of the set $A$ (see Section 2 for the definition). (The existence of $\varepsilon_d\in(0,\infty)$ when $d\ge5$ was proved in jain1968range; while o

Figures (1)

  • Figure 1: This figure illustrates the construction of ${\rm pp}(\eta;A,D)$. The boundaries of sets $A$ and $D$ are shown in black, and that of the neighborhood $B(D,L_n)$ is shown in orange. The blue curve indicates the trajectory $\eta$, where the union of the solid segments represent the proper part ${\rm pp}(\eta;A,D)$.

Theorems & Definitions (61)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 51 more