Chemical Distance in Random Structures

• Chemical distance is defined as the minimal path length constrained to remain within a random medium, serving as an intrinsic metric across diverse models.
• It exhibits varied scaling regimes—from linear and polylogarithmic to ultrasmall behaviors—depending on geometry, correlations, and tail properties of long connections.
• Applications span Bernoulli percolation, long-range graphs, Gaussian fields, and CLE carpets, informing shape theorems, large deviations, and connectivity analyses.

Chemical distance is the intrinsic distance induced by a random medium: it is the minimal length of a path constrained to remain in the random graph or random excursion set under consideration. In Bernoulli bond percolation on $\mathbb Z^d$, it is the graph distance $D^{G_p}(x,y)$ between vertices connected by open edges; in general spatial random graphs it is the minimal number of graph steps $D_G(x,y)$; in continuum excursion sets it is the Euclidean arclength of the shortest rectifiable path staying inside the set; and in conformal loop ensemble carpets it is approximated by minimizing the Lebesgue measure of $\epsilon$-neighborhoods of admissible carpet paths. Across these settings, chemical distance may be linearly comparable to Euclidean distance, polylogarithmic, or of order $\log\log$, depending on the geometry, correlation structure, and tail behavior of long connections (Dembin et al., 2022, Lüchtrath, 2024, Vernotte, 23 Apr 2026, Miller, 2021).

1. Definition and model-dependent variants

In supercritical bond percolation on $\mathbb Z^d$, the percolation graph is

$G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$

and the chemical distance is

$$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$

with the convention $D^{G_p}(x,y)=+\infty$ if $x$ and $D^{G_p}(x,y)$0 are not connected in $D^{G_p}(x,y)$1. Any path achieving the infimum is called a geodesic, and geodesics are necessarily self-avoiding. The same terminology appears on general transitive graphs, where one writes

$D^{G_p}(x,y)$2

and on spatial random graphs,

$D^{G_p}(x,y)$3

These are all discrete intrinsic metrics, although the ambient spaces and connectivity rules differ substantially (Dembin et al., 2022, Gorski et al., 12 Jul 2025, Lüchtrath, 2024).

In continuum models, the same idea is expressed through Euclidean path length. For a connected subset $D^{G_p}(x,y)$4, the chemical distance is

$D^{G_p}(x,y)$5

where $D^{G_p}(x,y)$6 is the family of continuous rectifiable paths in $D^{G_p}(x,y)$7 joining $D^{G_p}(x,y)$8 to $D^{G_p}(x,y)$9. In supercritical excursion sets of planar Gaussian fields one writes $D_G(x,y)$0 for the path metric inside $D_G(x,y)$1, while in higher-dimensional smooth Gaussian fields one writes $D_G(x,y)$2 or $D_G(x,y)$3 for the infimum of Euclidean arclengths of curves inside $D_G(x,y)$4. In simple CLE carpets, the approximation

$D_G(x,y)$5

uses $D_G(x,y)$6, the Lebesgue measure of the $D_G(x,y)$7-neighborhood of a carpet path $D_G(x,y)$8, as a surrogate for discrete chemical length (Vernotte, 2023, Vernotte, 28 Mar 2025, Miller, 2021).

This variability of definitions is not a matter of notation alone. It reflects the fact that chemical distance is always an intrinsic metric, but the ambient randomness may be combinatorial, geometric, correlated, oriented, or conformally invariant. A plausible implication is that statements about “linear” or “superlinear” chemical distance must always be interpreted relative to the native geometry of the model.

2. Supercritical percolation, time constants, and upper-tail mechanisms

For supercritical Bernoulli bond percolation on $D_G(x,y)$9, $\epsilon$0, there is almost surely a unique infinite open cluster $\epsilon$1, and for any $\epsilon$2 there exists a deterministic norm $\epsilon$3 such that

$\epsilon$4

Upper-tail deviations of the event

$\epsilon$5

were known to decay exponentially. In dimension $\epsilon$6, for $\epsilon$7 small enough, the upper-tail large deviation rate exists and is given by

$\epsilon$8

with

$\epsilon$9

The function $\log\log$0 is defined through space-time cut-point events $\log\log$1, is continuous, convex, positively homogeneous, and satisfies $\log\log$2 for $\log\log$3. The paper identifies the upper-tail mechanism as local obstruction near the endpoints: space-time cut-points force geodesics either to go in a non-optimal direction or to wiggle considerably. In $\log\log$4, box-to-box upper tails have surface-order decay $\log\log$5, which rules out global wall mechanisms at linear large-deviation speed; in $\log\log$6, by contrast, global and local mechanisms can compete (Dembin et al., 2022).

The same linear-growth picture extends beyond $\log\log$7. On any transitive graph of polynomial growth, for every $\log\log$8 there exist $\log\log$9 and $\mathbb Z^d$0 such that

$\mathbb Z^d$1

for all $\mathbb Z^d$2, all $\mathbb Z^d$3, and all $\mathbb Z^d$4. The same work defines a pseudometric

$\mathbb Z^d$5

where $\mathbb Z^d$6 is the closest vertex of the infinite cluster to $\mathbb Z^d$7, and proves Lipschitz continuity in $\mathbb Z^d$8 for normalized mean asymptotic distances. Along automorphism orbits $\mathbb Z^d$9, Kingman’s theorem gives time constants $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$0 via

$G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$1

This places chemical distance in supercritical percolation within the general framework of asymptotic norms and shape theorems, while also showing that the upper-tail geometry can be highly model-specific (Gorski et al., 12 Jul 2025).

A common misconception is that supercritical chemical-distance upper tails are always caused by global barriers. The cut-point analysis in $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$2 suggests that, at least for $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$3 and small upper deviations, the efficient mechanism is instead local and endpoint-driven.

3. Critical planar behavior and shortest-path exponents

At criticality in two-dimensional percolation, chemical distance ceases to be linearly comparable to Euclidean distance. For critical bond percolation on $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$4, the shortest left–right open crossing length $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$5 and the lowest crossing length $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$6 of a box $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$7 are distinct observables. Morrow–Zhang-type estimates place $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$8 at scale $G_p := \big(\mathbb Z^d, \{e\in E^d: B_e=1\}\big),$9, where $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$0 is the three-arm probability. Damron–Hanson–Sosoe proved that the shortest crossing is asymptotically much shorter than the lowest crossing: $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$1 in probability conditioned on the existence of a crossing, and $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$2. The later quantitative refinement established that for some $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$3,

$$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$4

and in the triangular-lattice case, where $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$5, this gives a strict upper bound below $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$6 for the chemical distance exponent of box crossings (Damron et al., 2015, Damron et al., 2017, Damron, 2016).

Critical point-to-point behavior is even less regular. For nearest neighbors $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$7 and $$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$8,

$$D^{G_p}(x,y) := \inf\big\{|r|:\ r \text{ is a path from } x \text{ to } y \text{ in } G_p\big\},$$9

and more generally $D^{G_p}(x,y)=+\infty$0 for small $D^{G_p}(x,y)=+\infty$1. At the same time, conditional upper bounds remain governed by the three-arm scale: if $D^{G_p}(x,y)=+\infty$2, then

$D^{G_p}(x,y)=+\infty$3

For point-to-surface distance in $D^{G_p}(x,y)=+\infty$4, the bound

$D^{G_p}(x,y)=+\infty$5

shows that the same multi-arm quantity controls radial and box-crossing distances. These results indicate heavy tails, strong non-concentration, and a shortest-path exponent that is not captured by the lowest crossing (Damron et al., 2016).

The planar random-cluster model at criticality exhibits an analogous phenomenon. For cluster weight $D^{G_p}(x,y)=+\infty$6, the expected shortest horizontal crossing length $D^{G_p}(x,y)=+\infty$7 satisfies

$D^{G_p}(x,y)=+\infty$8

extending the Bernoulli-percolation shortcut theory to dependent FK clusters. The proof requires a complete proof of the strong arm separation lemma for the random-cluster model and replaces the role of Reimer-type inequalities by domain Markov property, FKG, and FK-specific gluing estimates (Reeves, 2023).

A persistent misconception is that “chemical distance at criticality” is essentially the same as the length of a canonical extremal path such as the lowest crossing. The ratio results show that the shortest path has genuinely different scaling.

4. Long-range, ultrasmall, and Euclidean-comparison regimes

In long-range percolation, chemical distance can be dramatically smaller than any power of Euclidean distance. For long-range percolation on $D^{G_p}(x,y)=+\infty$9 and $x$0 with connection probabilities proportional to $x$1, $x$2, one has

$x$3

in probability on $x$4, where

$x$5

In the continuum model,

$x$6

for a positive continuous deterministic function $x$7 satisfying $x$8. Outside this intermediate regime, the same survey records that $x$9 corresponds to linear scaling and $D^{G_p}(x,y)$00 to an ultra-small-world regime with bounded limiting distance (Biskup et al., 2017).

A complementary line of work isolates conditions under which Euclidean shortcuts are impossible. For translation invariant, locally finite spatial random graphs on stationary point processes in $D^{G_p}(x,y)$01, if the graph has polynomial mixing

$D^{G_p}(x,y)$02

and no-long-edge estimates

$D^{G_p}(x,y)$03

then there exist $D^{G_p}(x,y)$04 and $D^{G_p}(x,y)$05 such that

$D^{G_p}(x,y)$06

This yields linear lower bounds on chemical distance relative to Euclidean distance, uniformly for one endpoint in a fixed inner box and the other far away. The result applies to models such as the weight-dependent random connection model, Boolean and soft Boolean models, interference models, and ellipses percolation (Lüchtrath, 2024).

In other geometric random graphs, the relevant scale is $D^{G_p}(x,y)$07. For geometric random graphs with long edges and scale-free degree distribution, the degree-tail parameter is $D^{G_p}(x,y)$08, while the spatial decay parameter is $D^{G_p}(x,y)$09. The sharp ultrasmall boundary is

$D^{G_p}(x,y)$10

In that regime, conditioning on connectivity,

$D^{G_p}(x,y)$11

with high probability, while if $D^{G_p}(x,y)$12 the graph is not ultrasmall and one has lower bounds of order $D^{G_p}(x,y)$13. The same constant $D^{G_p}(x,y)$14 appears across soft Boolean, age-dependent random connection, and reinforced age-dependent random connection models under the stated upper and lower kernel assumptions (Gracar et al., 2021).

The Poisson Boolean model with rotation-invariant convex bodies and regularly varying diameters also exhibits an explicit $D^{G_p}(x,y)$15-law. If $D^{G_p}(x,y)$16 for all $D^{G_p}(x,y)$17 and

$D^{G_p}(x,y)$18

then with

$D^{G_p}(x,y)$19

the chemical distance between far-apart connected points behaves as

$D^{G_p}(x,y)$20

If some $D^{G_p}(x,y)$21, the distance is $D^{G_p}(x,y)$22; if $D^{G_p}(x,y)$23 for all $D^{G_p}(x,y)$24, it is larger than $D^{G_p}(x,y)$25 for every $D^{G_p}(x,y)$26 (Gracar et al., 24 Mar 2025).

These results show that “small-world” behavior is not determined by heavy-tailed degrees alone. A plausible implication is that geometry and anisotropy can move the threshold for ultrasmallness even when the degree distribution remains scale-free.

5. Correlated Gaussian fields, Gaussian free fields, and continuum excursion sets

For level sets of the Gaussian free field on $D^{G_p}(x,y)$27, $D^{G_p}(x,y)$28, chemical distance in the percolating regime $D^{G_p}(x,y)$29 is near-linear but not yet at the Bernoulli scale of exponential tails. Writing $D^{G_p}(x,y)$30 for the graph distance inside $D^{G_p}(x,y)$31, one has

$D^{G_p}(x,y)$32

where $D^{G_p}(x,y)$33 is the set of vertices in components of $D^{G_p}(x,y)$34 with $D^{G_p}(x,y)$35-diameter $D^{G_p}(x,y)$36. There are matching lower bounds in the sense that for any $D^{G_p}(x,y)$37,

$D^{G_p}(x,y)$38

The proof uses the Gibbs–Markov decomposition, a renormalization scheme, and capacity-based bounds for harmonic averages (Peretz, 6 Jan 2025).

The two-dimensional discrete Gaussian free field exhibits a different phenomenon for the two-sided level set

$D^{G_p}(x,y)$39

For each $D^{G_p}(x,y)$40, there exists $D^{G_p}(x,y)$41 such that the probability that there exists a $D^{G_p}(x,y)$42-open path $D^{G_p}(x,y)$43 in $D^{G_p}(x,y)$44 with Euclidean span at least $D^{G_p}(x,y)$45 and length at most $D^{G_p}(x,y)$46 tends to $D^{G_p}(x,y)$47 as $D^{G_p}(x,y)$48. The paper gives the quantitative dependence $D^{G_p}(x,y)$49 for some absolute constant $D^{G_p}(x,y)$50, implying that the chemical distance exponent in this model is strictly larger than $D^{G_p}(x,y)$51 whenever macroscopic connections exist (Gao et al., 2020).

In planar Gaussian excursion sets, the continuum chemical distance is the Euclidean length of the shortest path inside $D^{G_p}(x,y)$52. For supercritical $D^{G_p}(x,y)$53, and under regularity, positivity, and decay assumptions on the covariance kernel, if $D^{G_p}(x,y)$54 and $D^{G_p}(x,y)$55 are connected in $D^{G_p}(x,y)$56, then with high probability

$D^{G_p}(x,y)$57

for any fixed $D^{G_p}(x,y)$58, with an explicit failure bound consisting of stretched-exponential and super-polynomial terms. The argument combines RSW-type crossing estimates, a discretized field $D^{G_p}(x,y)$59, and Kac–Rice bounds on the total boundary length of local components (Vernotte, 2023).

For smooth Gaussian fields in higher dimension, with excursion set $D^{G_p}(x,y)$60, the main theorem is proved for $D^{G_p}(x,y)$61. There exists $D^{G_p}(x,y)$62 such that

$D^{G_p}(x,y)$63

where

$D^{G_p}(x,y)$64

The proof uses finite-range approximation, stochastic domination by high-parameter Bernoulli site percolation on a renormalized lattice, local uniqueness events, and local implicit-function control of the excursion-set geometry (Vernotte, 28 Mar 2025).

The continuum “shadow” model based on the slope field

$D^{G_p}(x,y)$65

gives another correlated setting. For Lebesgue-almost-all supercritical levels $D^{G_p}(x,y)$66, there exists $D^{G_p}(x,y)$67 such that

$D^{G_p}(x,y)$68

for every $D^{G_p}(x,y)$69. The proof is explicitly “in the spirit of the Antal–Pisztora theorem” but the tail is polynomial because the local control relies on Kac–Rice estimates for level-set length rather than purely discrete renormalization (Vernotte, 23 Apr 2026).

A common misconception is that positive association or supercriticality alone should force Bernoulli-type linear bounds with exponential tails. The Gaussian and GFF examples show that correlations, continuum geometry, and local regularity can substantially weaken the available chemical-distance estimates.

6. Other geometries, limiting laws, and open directions

Chemical distance also appears in oriented, interlacement-type, and scaling-limit settings. In the half-orthant model on $D^{G_p}(x,y)$70, the distance $D^{G_p}(x,y)$71 is the minimal number of consistent directed steps from $D^{G_p}(x,y)$72 to $D^{G_p}(x,y)$73. In the regimes $D^{G_p}(x,y)$74 or $D^{G_p}(x,y)$75, Kingman–Liggett subadditivity yields deterministic directional time constants $D^{G_p}(x,y)$76 with

$D^{G_p}(x,y)$77

almost surely and in $D^{G_p}(x,y)$78. The associated limit shape

$D^{G_p}(x,y)$79

is convex, with flat facets in the positive orthant and in further directions coming from oriented percolation cones, while other directions satisfy $D^{G_p}(x,y)$80 (Beaton et al., 2024).

For sufficiently supercritical finitary random interlacements $D^{G_p}(x,y)$81 in $D^{G_p}(x,y)$82, $D^{G_p}(x,y)$83, there is a unique infinite cluster $D^{G_p}(x,y)$84, and for large $D^{G_p}(x,y)$85 one has

$D^{G_p}(x,y)$86

where $D^{G_p}(x,y)$87 is the chemical distance and $D^{G_p}(x,y)$88 is the $D^{G_p}(x,y)$89-norm. This implies a shape theorem for intrinsic balls and a local uniqueness property for large clusters in boxes. The proof constructs a “highway system” subcluster $D^{G_p}(x,y)$90 by multi-scale renormalization and capacity estimates for lucky path segments (Cai et al., 2020).

At high-dimensional criticality, the natural scale is diffusive rather than linear. For critical Bernoulli percolation on $D^{G_p}(x,y)$91 with $D^{G_p}(x,y)$92 in the nearest-neighbor model, or spread-out percolation for $D^{G_p}(x,y)$93, conditioning on $D^{G_p}(x,y)$94, the rescaled chemical distance converges in distribution: $D^{G_p}(x,y)$95 The limit $D^{G_p}(x,y)$96 has density

$D^{G_p}(x,y)$97

and coincides with the hitting time of a Brownian motion in $D^{G_p}(x,y)$98 conditioned to hit a fixed unit vector. The same universal limit law applies to the effective resistance and the number of pivotal edges on the long connection (Chatterjee et al., 7 Sep 2025).

At the opposite end of the scaling hierarchy, the CLE$D^{G_p}(x,y)$99 carpet, $D_G(x,y)$00, admits a tight family of continuum approximations to chemical distance. If $D_G(x,y)$01 is the carpet and

$D_G(x,y)$02

with $D_G(x,y)$03 the Lebesgue measure of the $D_G(x,y)$04-neighborhood of $D_G(x,y)$05, then

$D_G(x,y)$06

is tight, where $D_G(x,y)$07 is the median of $D_G(x,y)$08. Any subsequential limit is a geodesic metric on $D_G(x,y)$09 and is Hölder continuous with respect to the Euclidean metric. The paper conjectures uniqueness of the subsequential limit, conformal covariance, and convergence of discrete chemical distances in loop models to this CLE metric (Miller, 2021).

Across these works, several open problems recur. The supercritical percolation large-deviation theory is proved only for small $D_G(x,y)$10 in the cut-point regime, leaving the full rate function open (Dembin et al., 2022). In critical planar percolation and FK models, the exact shortest-path exponent remains unknown despite strict improvements over the lowest-path scale (Damron et al., 2017, Reeves, 2023). For GFF level sets and Gaussian excursion sets, a shape theorem or deterministic time constant analogous to Bernoulli percolation remains open (Peretz, 6 Jan 2025, Vernotte, 28 Mar 2025). In CLE carpets, uniqueness and conformal covariance of the continuum chemical metric are conjectural (Miller, 2021). These open directions suggest that chemical distance is best understood not as a single metric phenomenon, but as a family of intrinsic geometries whose scaling is controlled by phase, dimension, correlation, and the availability of long connections.