Random Interlacements Percolation

• Random interlacements percolation is a framework that models connectivity in ℤᵈ using a Poisson cloud of doubly-infinite trajectories and occupation-time level sets.
• The approach leverages an exact Gaussian free field isomorphism to translate occupation-time events into level-set conditions, yielding sharp connectivity bounds.
• Renormalization, decoupling, and sparse tree embeddings facilitate stretched-exponential decay estimates and support a robust two-parameter phase diagram.

Searching arXiv for recent and foundational papers on random interlacements percolation. Found relevant papers including the foundational work on level-set percolation for random interlacements and the Gaussian free field (Rodriguez, 2013), sharpness and threshold equality for the vacant set (Duminil-Copin et al., 2023), local uniqueness in the small-intensity regime (Drewitz et al., 2012), decoupling on $G\times \mathbb{Z}$ (Sznitman, 2010), and recent sharp connectivity bounds (Goswami et al., 3 Apr 2025). Percolation of random interlacements concerns connectivity properties of random subsets of $\mathbb{Z}^d$, $d \ge 3$, generated by a Poisson cloud of doubly-infinite nearest-neighbor trajectories. In the continuous-time framework, the basic objects are not only the classical interlacement set $\mathcal{I}^u$ and vacant set $\mathcal{V}^u$ at level $u$, but also occupation-time level sets obtained by thresholding the local-time field $L_{x,u}$. This yields a two-parameter family of occupied and empty phases whose percolative behavior can be analyzed through an exact isomorphism with the Gaussian free field, together with renormalization and decoupling methods (Rodriguez, 2013).

1. Continuous-time interlacements and occupation-time level sets

Continuous-time random interlacements on $\mathbb{Z}^d$, $d \ge 3$, are defined as a Poisson point process on the space of doubly-infinite nearest-neighbor trajectories modulo time-shift, endowed with i.i.d. exponential holding times of parameter $1$ on each discrete step. The intensity measure is built from potential theory through the equilibrium measure $\mathbb{Z}^d$0 and the capacity $\mathbb{Z}^d$1 of finite sets. At level $\mathbb{Z}^d$2, the interlacement set is

$\mathbb{Z}^d$3

and the vacant set is

$\mathbb{Z}^d$4

The occupation-time field is

$\mathbb{Z}^d$5

the total continuous time spent at $\mathbb{Z}^d$6 by all interlacement trajectories with label at most $\mathbb{Z}^d$7 (Rodriguez, 2013).

Thresholding $\mathbb{Z}^d$8 produces the occupied and empty level sets

$\mathbb{Z}^d$9

for $d \ge 3$0. The classical model is recovered at $d \ge 3$1: $d \ge 3$2 Thus $d \ge 3$3 for all $d \ge 3$4, whereas $d \ge 3$5. Increasing $d \ge 3$6 shrinks the occupied phase and enlarges the empty phase. The geometric interpretation is that time-thresholding removes lightly visited sites from the interlacement cluster and can therefore destroy long-range connectivity even though the underlying trajectory cloud remains extensive.

2. Gaussian free field isomorphism and level-set translation

A central structural input is Sznitman’s isomorphism theorem. Let $d \ge 3$7 be the centered Gaussian free field with covariance

$d \ge 3$8

where $d \ge 3$9 is the Green function of simple random walk. Then for every $\mathcal{I}^u$0,

$\mathcal{I}^u$1

has the same law as

$\mathcal{I}^u$2

Equivalently, for any finite $\mathcal{I}^u$3,

$\mathcal{I}^u$4

This is an exact identity in law, not an independent coupling, but it permits direct comparison of occupation-time exceedance events with Gaussian free field level events (Rodriguez, 2013).

For the occupied phase, the identity yields

$\mathcal{I}^u$5

Accordingly, for $\mathcal{I}^u$6, the law $\mathcal{I}^u$7 of $\mathcal{I}^u$8 is stochastically dominated by the law of the two-sided Gaussian free field level set

$\mathcal{I}^u$9

For the empty phase, the same identity shows that the constraint $\mathcal{V}^u$0 can only occur if either $\mathcal{V}^u$1 is sufficiently negative to counteract the shift $\mathcal{V}^u$2, or $\mathcal{V}^u$3 is itself large. This leads to upper bounds on connectivity in $\mathcal{V}^u$4 when $\mathcal{V}^u$5 is large relative to $\mathcal{V}^u$6.

The Gaussian free field comparison relies on two classes of level sets. The two-sided level set is

$\mathcal{V}^u$7

with critical threshold

$\mathcal{V}^u$8

One has $\mathcal{V}^u$9 for all $u$0, and for sufficiently large $u$1,

$u$2

The one-sided level set

$u$3

has its own threshold $u$4, finite for all $u$5 and strictly positive in high dimensions.

3. Renormalization, sparse trees, and decoupling

The proof architecture is multiscale. One fixes scales

$u$6

and boxes

$u$7

A sparse embedding of a dyadic tree into $u$8 produces $u$9 well-separated descendant boxes inside a parent box. This geometry allows one to define recursively propagating bad events and to quantify their decay through scale (Rodriguez, 2013).

For localized events $L_{x,u}$0 depending only on coordinates in $L_{x,u}$1, one introduces

$L_{x,u}$2

where $L_{x,u}$3. The decoupling inequality has the form

$L_{x,u}$4

with $L_{x,u}$5, and the sprinkling increment $L_{x,u}$6 proportional to

$L_{x,u}$7

Choosing $L_{x,u}$8 appropriately propagates the recursion and yields stretched-exponential bounds. The resulting exponent is

$L_{x,u}$9

The geometric counterpart is the cascading principle for bad blocks. One declares a level-$\mathbb{Z}^d$0 box bad when it contains a local obstruction, and defines a level-$\mathbb{Z}^d$1 bad event when $\mathbb{Z}^d$2 well-separated bad level-$\mathbb{Z}^d$3 boxes are embedded inside the corresponding $\mathbb{Z}^d$4-box. If

$\mathbb{Z}^d$5

for all $\mathbb{Z}^d$6 and $\mathbb{Z}^d$7, then long connections through bad boxes satisfy the same doubly exponential scale estimate, which interpolates into the stretched-exponential annulus-crossing bounds used throughout the theory. This machinery is used both for Gaussian free field level sets and, via the isomorphism, for occupation-time level sets of interlacements.

4. Critical parameters for occupied and empty phases

The occupation-time model introduces two natural critical families. For fixed $\mathbb{Z}^d$8, the occupied-set threshold is

$\mathbb{Z}^d$9

and for fixed $d \ge 3$0, the empty-set threshold is

$d \ge 3$1

Before the summary table, two structural facts are essential. First, for $d \ge 3$2 and $d \ge 3$3, the law $d \ge 3$4 of $d \ge 3$5 is translation invariant and satisfies the finite energy property

$d \ge 3$6

Burton–Keane then yields uniqueness of the infinite cluster in the supercritical regime; an analogous statement holds for the empty-phase law. Second, both the occupied and empty phases exhibit stretched-exponential decay of crossing probabilities in their respective subcritical regimes (Rodriguez, 2013).

Random set	Critical parameter	Established behavior
$d \ge 3$7	$d \ge 3$8	$d \ge 3$9 for $1$0; non-decreasing in $1$1; $1$2
$1$3	$1$4	$1$5; $1$6 for all $1$7; non-decreasing in $1$8; $1$9

For the occupied phase, positivity of $\mathbb{Z}^d$00 for every $\mathbb{Z}^d$01 is proved by comparing $\mathbb{Z}^d$02 with an independent Bernoulli thinning of $\mathbb{Z}^d$03 obtained from first-passage holding times of minimal-label trajectories. This yields slab percolation for sufficiently small $\mathbb{Z}^d$04: for each fixed $\mathbb{Z}^d$05, there exist $\mathbb{Z}^d$06 and $\mathbb{Z}^d$07 such that $\mathbb{Z}^d$08 percolates almost surely in the slab $\mathbb{Z}^d$09 for all $\mathbb{Z}^d$10. Finiteness of $\mathbb{Z}^d$11 follows from Gaussian domination: if

$\mathbb{Z}^d$12

then

$\mathbb{Z}^d$13

For the empty phase, $\mathbb{Z}^d$14 is finite for every $\mathbb{Z}^d$15. The argument constructs $\mathbb{Z}^d$16-bad boxes through three local obstructions: a large positive GFF excursion, a large negative GFF excursion, or an unusually small value of

$\mathbb{Z}^d$17

inside the box. When $\mathbb{Z}^d$18 is sufficiently large relative to $\mathbb{Z}^d$19, these bad events are rare at all scales, and one obtains

$\mathbb{Z}^d$20

for all $\mathbb{Z}^d$21. The thresholds $\mathbb{Z}^d$22 and $\mathbb{Z}^d$23 therefore describe a genuine two-parameter phase diagram rather than a trivial deformation of the $\mathbb{Z}^d$24 model.

5. Classical vacant-set percolation and later sharpness results

The occupation-time theory sits on top of the classical interlacement geometry. For the unthresholded model, the interlacement set $\mathbb{Z}^d$25 is almost surely connected for every $\mathbb{Z}^d$26, whereas the vacant set $\mathbb{Z}^d$27 undergoes a non-trivial phase transition at a threshold $\mathbb{Z}^d$28 (Sapozhnikov, 2014). The $\mathbb{Z}^d$29 slice of the occupation-time model is therefore already highly nontrivial on the vacant side, and the thresholded model extends this by allowing one to study under-visited and over-visited regions simultaneously.

In the low-intensity regime, local uniqueness for the vacant set was established in all dimensions $\mathbb{Z}^d$30: for sufficiently small $\mathbb{Z}^d$31, a large box contains, with stretched-exponentially high probability, a unique macroscopic component of $\mathbb{Z}^d$32, and finite vacant clusters have stretched-exponential tail bounds for diameter and volume (Drewitz et al., 2012). This local picture is exactly the sort of strongly percolative behavior that later became formalized in the threshold $\mathbb{Z}^d$33.

A major later development is the sharpness theory for the classical vacant set. The phase transition is sharp: for $\mathbb{Z}^d$34, one has stretched-exponential decay of the two-point function, and in $\mathbb{Z}^d$35 the decay can be made exponential; for $\mathbb{Z}^d$36, one has robust existence and local uniqueness events with stretched-exponential error bounds. Moreover,

$\mathbb{Z}^d$37

for all $\mathbb{Z}^d$38, so the global threshold, the strong percolation threshold, and the subcritical crossing-decay threshold coincide (Duminil-Copin et al., 2023). Relative to the occupation-time theory, this resolves for $\mathbb{Z}^d$39 the threshold-coincidence issue that remained open in the original two-parameter setting.

Recent connectivity results sharpen the subcritical and supercritical picture even further. For the truncated two-point function of the vacant set,

$\mathbb{Z}^d$40

one now has in dimension three

$\mathbb{Z}^d$41

and in dimensions $\mathbb{Z}^d$42, for $\mathbb{Z}^d$43,

$\mathbb{Z}^d$44

for suitable $\mathbb{Z}^d$45 (Goswami et al., 3 Apr 2025). In dimension three the rate depends on $\mathbb{Z}^d$46 only through its Euclidean norm at principal exponential order, which offers a precise asymptotic counterpart to the earlier stretched-exponential bounds.

6. Extensions, limitations, and open directions

Several extensions show that the percolation theory of random interlacements is not confined to the Euclidean lattice. On weighted graphs of the form $\mathbb{Z}^d$47, under $\mathbb{Z}^d$48-Ahlfors regularity and sub-Gaussian heat-kernel assumptions parameterized by $\mathbb{Z}^d$49, decoupling inequalities analogous to the lattice case yield

$\mathbb{Z}^d$50

as well as stretched-exponential bounds for annular crossing probabilities in the vacant set above $\mathbb{Z}^d$51. In the regime $\mathbb{Z}^d$52, one further has positivity of the half-plane threshold $\mathbb{Z}^d$53, hence $\mathbb{Z}^d$54 (Sznitman, 2010). This suggests that the renormalization philosophy underlying interlacement percolation is robust under substantial geometric inhomogeneity.

The model is also stable under small quenched perturbations. If each occupied site of $\mathbb{Z}^d$55 is flipped to vacant, and each vacant site to occupied, independently with probability $\mathbb{Z}^d$56, then for every $\mathbb{Z}^d$57 and $\mathbb{Z}^d$58, the perturbed interlacement still percolates almost surely for sufficiently small $\mathbb{Z}^d$59. On the vacant side, there remains a non-trivial noisy critical threshold $\mathbb{Z}^d$60, and

$\mathbb{Z}^d$61

for small noise (Rath et al., 2011). A plausible implication is that the large-scale geometry of the interlacement and vacant phases is stable under weak independent disorder, despite the underlying long-range correlations.

For the occupation-time level sets of $\mathbb{Z}^d$62 and $\mathbb{Z}^d$63, several issues remain open (Rodriguez, 2013). It is not known whether the auxiliary thresholds defined through stretched-exponential crossing decay satisfy

$\mathbb{Z}^d$64

The exact values of $\mathbb{Z}^d$65, $\mathbb{Z}^d$66, the Gaussian comparison threshold $\mathbb{Z}^d$67, and the decay exponent $\mathbb{Z}^d$68 are not determined. For the two-sided Gaussian free field threshold $\mathbb{Z}^d$69, finiteness is known, but the exact behavior near criticality, uniqueness of the infinite cluster at low levels, and critical exponents remain unresolved. Continuity of the percolation probability in the two-parameter occupation-time model is likewise not addressed there.

These unresolved questions define the current frontier of the subject. The established theory shows that occupation-time thresholding turns random interlacements into a genuinely two-parameter correlated percolation model, with exact Gaussian free field comparison, nontrivial occupied and empty phases, and renormalization-controlled subcritical decay. Later sharpness results for the classical vacant set strongly suggest that a comparably sharp description of the full $\mathbb{Z}^d$70-phase diagram should exist, but at present it remains only partially charted.