Sharp connectivity bounds for the vacant set of random interlacements

TL;DR

The paper analyzes the vacant set of random interlacements on $\mathbb{Z}^d$ ($d\ge 3$), deriving sharp lower bounds for the truncated two-point function $\tau_{u}^{\mathrm{tr}}(x)$ across all $u>0$, with sharpness up to the principal exponential order in $d=3$ and for $u\neq u_*$ in $d\ge 4$. The authors introduce a spatial tilting profile $f$ and construct a tilted interlacement measure $\widetilde{\mathbb{P}}_{f}$ to create oblique corridors that bias trajectories in a controlled, near-critical fashion, while maintaining a tractable entropy cost. A general lower-bound framework (LB-gen) for oblique connection–disconnection events is developed, and is combined with a robust coupling of tilted and untitled interlacements on local boxes (via soft local times and entrance-law comparisons) to transfer tilt-typical events to the original measure. The work yields a rotationally invariant, Euclidean-distance-based lower bound in $d=3$ and matching exponential-order lower bounds in higher dimensions, thereby completing sharp asymptotics in this percolation model and offering new tools for handling non-monotone, nonlocal percolation phenomena. The methods hinge on precise hitting-probability estimates for oblique sets and meticulous control of relative entropy, enabling a deep understanding of the scaling limit behavior near criticality.

Abstract

We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal exponential order for all $u$ in dimension three and all $u \neq u_\ast$ in higher dimensions, where $u_*$ refers to the critical parameter of the model, and they match the upper bounds derived in the article arXiv:2503.14497. In dimension three, our results further imply that the truncated two-point function grows at large distances $x$ at a rate that depends on $x$ only through its Euclidean norm, which offers a glimpse of the expected (Euclidean) invariance of the scaling limit at criticality. The rate function is atypical, it incurs a logarithmic correction and comes with an explicit pre-factor that converges to $0$ as the parameter $u$ approaches the critical point $u_*$ from either side. A particular challenge stems from the combined effects of lack of monotonicity due to the truncation in the super-critical phase, and the precise (rotationally invariant) controls we seek, that measure the effects of a certain "harmonic humpback" function. Among others, their derivation relies on rather fine estimates for hitting probabilities of the random walk in arbitrary direction $e$, which witness this invariance at the discrete level, and preclude straightforward applications of projection arguments.

Figures (2)

• Figure 1: The "harmonic humpback" function $f$. The purple part is joint to both the red and blue curve, which distinguish sub- and super-critical regimes. In reality, each curve depicts a section orthogonal to the direction $e \in S^{d-1}$, which is arbitrary. The plateaux correspond to spatial regions at which trajectories arrive at roughly constant (but suitably modulated) intensity. For practical purposes the reader should think of $v \approx u$.
• Figure 2: A configuration of cylinders in a generic direction $\bm{e = {x}/{|x|}}$. We illustrate a sample path contributing to the martingale $M_t$ introduced in \ref{['eq:hit-1']}. The point $z$ varies over all of $\widetilde{T}$, which includes the two 'stablilizing' rods (dashed). These are used to achieve the desired uniformity for the lower bound in \ref{['eq:mart-2']}. In reality, all lines and rectangles drawn are replaced by lattice approximations, cf. \ref{['eq:T_x']}-\ref{['eq:T_i']}.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• proof
• Lemma 2.2: $d = 3$
• proof : Proof of Lemma \ref{['C:mart']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['T:LB-RI']}
• Remark 3.2: one-arm estimates
• Proposition 4.1