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First passage percolation, local uniqueness for interlacements and capacity of random walk

Alexis Prévost

TL;DR

The paper addresses first passage percolation for the random interlacements model on transient graphs with polynomial volume growth and Green-function decay, establishing an asymptotically sharp lower bound for the probability that the FPP distance is sublinear and revealing near-critical behavior in the low-intensity regime. It develops a robust FPP framework on general graphs via coarse-graining, decoupling of interlacement excursions, soft local times, and entropy controls, culminating in a main FPP theorem that yields explicit large-deviation rates depending on the decay exponent $\nu$ of the Green function. These FPP bounds are then translated into sharp near-critical results for local uniqueness of random interlacements and capacity bounds for random walks in balls, with concrete dimension-specific refinements in $d=3$ and $d=4$ and a typical length scale $u^{-1/(d-2)}$ in the low-intensity regime. Overall, the work broadens the scope of FPP-type techniques from lattices to a versatile class of graphs, improving understanding of percolation, capacity, and geometric structure of interlacement-driven processes in subcritical and near-critical regimes.

Abstract

The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.

First passage percolation, local uniqueness for interlacements and capacity of random walk

TL;DR

The paper addresses first passage percolation for the random interlacements model on transient graphs with polynomial volume growth and Green-function decay, establishing an asymptotically sharp lower bound for the probability that the FPP distance is sublinear and revealing near-critical behavior in the low-intensity regime. It develops a robust FPP framework on general graphs via coarse-graining, decoupling of interlacement excursions, soft local times, and entropy controls, culminating in a main FPP theorem that yields explicit large-deviation rates depending on the decay exponent of the Green function. These FPP bounds are then translated into sharp near-critical results for local uniqueness of random interlacements and capacity bounds for random walks in balls, with concrete dimension-specific refinements in and and a typical length scale in the low-intensity regime. Overall, the work broadens the scope of FPP-type techniques from lattices to a versatile class of graphs, improving understanding of percolation, capacity, and geometric structure of interlacement-driven processes in subcritical and near-critical regimes.

Abstract

The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on , , the FPP distance is comparable to the graph distance with high probability. In this article, we give an asymptotically sharp lower bound on this last probability, which additionally holds on a large class of transient graphs with polynomial volume growth and polynomial decay of the Green function. When considering the interlacement set in the low-intensity regime, the previous bound is in fact valid throughout the near-critical phase. In low dimension, we also present two applications of this FPP result: sharp large deviation bounds on local uniqueness of random interlacements, and on the capacity of a random walk in a ball.
Paper Structure (18 sections, 24 theorems, 232 equations, 2 figures)

This paper contains 18 sections, 24 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local uniqueness for random interlacements
  3. Capacity of random walk
  4. Capacity of random walk
  5. First passage percolation
  6. Sketch of the proofs
  7. Outline of the rest of the article
  8. Preliminaries
  9. First passage percolation
  10. Decoupling of interlacements and coarse-graining
  11. A priori estimate
  12. Proof of Theorem \ref{['the:FPP']}
  13. Proof of the three intermediate propositions
  14. Proof of Proposition \ref{['pro:thm4.2disco']}
  15. Proof of Proposition \ref{['pro:softlocaltimes']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix some $\eta\in{(0,1)}$. There exist constants $0<c, C<\infty$, depending on $\eta$, such that for all $N\in{\mathbb{N}}$ and $u\in{(0,1/2]}$, if $G=\mathbb{Z}^3$ and $uN\geq C$ whereas if $G=\mathbb{Z}^4$ and $\sqrt{u}N\geq C\log(1/u)$

Figures (2)

  • Figure 1: Illustration of the proper embedding generated by a path $\gamma$ from $C_x^{l_k}$ to $\widetilde{C}_x^{L_k}$ (in red). Normal lines represent $C_{\cdot}^{l_n}$, dashed lines $\widetilde{C}_{\cdot}^{L_n}$, and dotted lines ${C}_{\cdot}^{L_n}$, with $n=k$ in black, $n=k-1$ in green, and $n=k-2$ in blue.
  • Figure 2: The strategy to ensure that the local uniqueness event does not occur: a) There is exactly one trajectory hitting $B(x'_{P'},N'/P)$, and this trajectory visits $B(z,N'/P)$ while staying confined in the tube $\widetilde{\mathcal{L}}^{P,N'}_z$. b) There are trajectories hitting $B(x'_M,N'/P)$, the first such trajectory visits $B(x,N'/P)$ while staying confined in the tube $\widetilde{\mathcal{L}}^{P,N'}_x$, whereas all the other such (dashed) trajectories never visit $B(x,\xi N)$. c) Any other (dashed) trajectory never visits $\widetilde{\mathcal{L}}_z^{P,N'}$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 50 more