Percolation Crossover Model

• Percolation crossover models are frameworks that describe how critical behaviors shift between distinct connectivity, transport, and universality classes as control parameters, length scales, or protocols vary.
• They elucidate transitions such as discontinuous to smooth conductivity and sub-diffusive to classical diffusion through finite-size scaling and regime-dependent observables.
• They also integrate atomistic ordering effects in alloys to map passivation thresholds, offering practical insights for designing corrosion-resistant materials.

Percolation crossover models describe situations in which a percolative system does not remain in a single asymptotic regime, but instead passes between distinct connectivity, transport, geometric, or universality-class behaviors as a control parameter, length scale, interaction range, disorder amplitude, or observation protocol is varied. In the literature, such crossovers include the transition from a drastically to a smoothly increasing conductivity in discontinuous percolation, from initial sub-diffusion to final classical diffusion on percolation lattices, from mean-field to $2d$ Directed Percolation in spatially embedded networks, and from $2D$-like to $3D$-like connectivity in thin films of chemically ordered alloys (Kim et al., 2014, Ghaemi et al., 2010, Santos et al., 2018, Roy et al., 2024). The same terminology also denotes a specific passivation framework for face centered cubic binary alloys with chemical short-range order (SRO), where the percolation threshold and thin-film crossover thickness become functions of the Warren-Cowley SRO parameter, thereby linking atomistic ordering to protective oxide formation (Roy et al., 2024).

1. Conceptual scope and scaling structure

At the most general level, a percolation crossover model formalizes the fact that critical behavior can be regime-dependent rather than globally described by a single exponent set. A central language for this is finite-size scaling (FSS), written as

$Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$

with $t$ the distance to criticality, $L$ the linear size, $\nu$ the correlation-length exponent, and $Y_Q$ an observable-dependent FSS exponent (Li et al., 2024). The crossover regime arises when one studies the asymptotic behavior of $\tilde Q(x)$ for $|x|\to\infty$ while $2D$0, or when the critical point is approached at a slower speed $2D$1 with $2D$2, leading to $2D$3-dependent scaling (Li et al., 2024).

This scaling viewpoint recurs across otherwise dissimilar models. In diffusion on a $2D$4 site-percolation lattice, the correlation length obeys

$2D$5

and the crossover from anomalous diffusion to classical diffusion is expected when the walk probes scales larger than $2D$6 (Ghaemi et al., 2010). In thin-film passivation models, the crossover from $2D$7-like to $2D$8-like percolation is expressed through a thickness-dependent threshold relation rather than through a conventional bulk FSS ansatz (Roy et al., 2024). In event-based ensembles for explosive and high-dimensional percolation, crossover FSS clarifies why pseudocritical and infinite-system critical points can exhibit different apparent exponents (Li et al., 2024).

A concise classification is therefore possible.

Setting	Control variable or scale	Reported crossover
Discontinuous percolation conductivity	$2D$9 and $3D$0	drastic to smooth conductivity growth (Kim et al., 2014)
Diffusion on percolation lattices	time or distance vs. $3D$1	sub-diffusion to classical diffusion (Ghaemi et al., 2010)
Contact process on embedded networks	long-range exponent $3D$2	mean-field to $3D$3 Directed Percolation (Santos et al., 2018)
FCC alloy passivation	SRO parameter and film thickness	$3D$4-$3D$5 percolation crossover (Roy et al., 2024)
Long-range $3D$6 percolation	decay exponent $3D$7	short-range to long-range universality at $3D$8 (Liu et al., 22 Sep 2025)

2. Transport and conductivity crossover phenomena

Transport observables provide some of the clearest realizations of percolation crossover. In the Spanning Cluster Avoiding (SCA) model, which exhibits a discontinuous percolation transition under a suppressive external bias, the conductivity $3D$9 is zero at the threshold $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$0 because the system is split into two large disconnected clusters separated by a fractal set of bridge bonds; immediately after $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$1, occupation of bridge bonds causes conductivity to increase drastically from zero; further from $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$2, the increase becomes smooth and resembles standard percolation (Kim et al., 2014). The effective-medium description is

$Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$3

and near threshold

$Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$4

with $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$5 and $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$6 (Kim et al., 2014). This model attributes the crossover to bridge-bond bottlenecks that short-circuit two already dense clusters.

Diffusive transport shows an analogous but observation-sensitive crossover. In lattice gas automata simulations of random walks on $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$7 site-percolation lattices, the mean-square displacement obeys

$Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$8

with $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/\nu}),$9 in the sub-diffusive regime and $t$0 in classical diffusion (Ghaemi et al., 2010). Individual configurations can show clear crossover times $t$1 from sub-diffusion to classical diffusion, but ensemble averaging over $t$2 configurations removes any manifestation of such crossovers, leaving an almost constant effective exponent. This establishes configuration dependence, not merely finite-size rounding, as a central methodological issue (Ghaemi et al., 2010).

Quantum transport adds a further layer. In a $t$3 quantum percolation model with dephasing introduced through Büttiker’s virtual probes, conductance evolves from quantum localization to classical Ohmic behavior as dephasing increases (Qi et al., 2019). An intermediate regime appears in which $t$4 increases with $t$5, producing an unexpected metallic phase before the fully classical limit; the scaling plot indicates a metal-insulator crossover rather than a sharp transition (Qi et al., 2019). A different transport crossover occurs in conductor-insulator composites, where the ratio $t$6 controls the transition from lattice-like percolation to tunneling-like hopping; for $t$7, conductivity has tunneling-like behavior independent of the specific microstructure (Ambrosetti et al., 2010).

3. Universality-class crossovers

Many percolation crossover models are best understood as flows between fixed points. In the contact process on spatially embedded networks with one long-range link per node and

$t$8

Monte Carlo simulations and finite-size scaling reveal mean-field critical exponents for $t$9, $L$0 directed-percolation exponents for $L$1, and continuously varying exponents in the crossover region $L$2 (Santos et al., 2018). Here the geometry of long-range connectivity tunes the effective dimensionality.

A related but symmetry-based crossover occurs in biased directed percolation, where

$L$3

The isotropic point is $L$4, while standard directed percolation is recovered at $L$5 (Zhou et al., 2011). Extensive simulations show that any asymmetry $L$6 is relevant at the isotropic percolation fixed point, and the system crosses over to directed-percolation exponents. Near the isotropic point,

$L$7

with the crossover exponent determined by the asymmetric scaling field (Zhou et al., 2011).

Disease-spreading models provide a dynamical counterpart. In the $L$8 SIRS model, finite recovery time $L$9 drives a crossover from the dynamical percolation class of SIR at $\nu$0 to the directed percolation class of SIS for any finite $\nu$1 (Saif, 2023). The phase boundary obeys

$\nu$2

with $\nu$3, where $\nu$4 in the simulations (Saif, 2023). This is a sharp statement about universality: the DyP fixed point is attained only at the singular limit of permanent immunity.

Long-range percolation makes the universality issue controversial. Large-scale simulations of a $\nu$5 long-range bond percolation model with occupation probabilities decaying as $\nu$6 report a crossover from short-range to long-range universality at $\nu$7, together with a pronounced jump in universal values and critical exponents at that point, explicitly described as being in contradiction to Sak’s criterion (Liu et al., 22 Sep 2025). By contrast, equivalent-neighbor bond percolation in $\nu$8 shows no evidence of a tricritical point separating mean-field and short-range behavior: all finite interaction ranges ultimately belong to the short-range universality class, while the mean-field limit is destabilized by a finite-range perturbation with renormalization exponent $\nu$9 (Ouyang et al., 2018).

4. Geometric, dimensional, and disorder-induced crossover

Not all percolation crossover models are primarily about transport or universality classes; many concern a change in geometry. Loop-erased random walk on percolation clusters crosses over from Euclidean to fractal geometry as the occupation probability decreases from $Y_Q$0 to $Y_Q$1 (Daryaei et al., 2013). In $Y_Q$2, the mean length satisfies

$Y_Q$3

near $Y_Q$4, and

$Y_Q$5

in the Euclidean regime, with $Y_Q$6, $Y_Q$7, $Y_Q$8, $Y_Q$9, and

$\tilde Q(x)$0

(Daryaei et al., 2013). At criticality the reported fractal dimension is $\tilde Q(x)$1 in $\tilde Q(x)$2.

First-passage percolation under strong disorder presents a crossover from bond-percolation universality to Kardar-Parisi-Zhang universality (Villarrubia et al., 2019). The mapping

$\tilde Q(x)$3

connects passage times to bond-occupation probabilities, and a new crossover length $\tilde Q(x)$4 determines the scale below which the model is described by bond-percolation criticality (Villarrubia et al., 2019). The interplay between $\tilde Q(x)$5 and the percolation correlation length $\tilde Q(x)$6 controls the transition from initial percolation-like growth to asymptotic KPZ scaling.

Temporal crossover can also occur within a single invasion process. In invasion percolation on the square lattice, autocorrelation functions of the external frontier length, radius, and roughness change from power-law decay at small times to exponential decay at long times, while near the crossover time they are fitted by log-normal functions (Tizdast et al., 2020). The increments of these quantities undergo an anticorrelation/correlation transition at the same time. This suggests that crossover may involve a genuine change in the effective noise structure of the growth dynamics rather than only a change in static connectivity.

5. The passivation percolation crossover model in binary alloys

In its most specific usage, the percolation crossover model is the framework developed for face centered cubic binary alloys with chemical short-range order to understand electrochemical passivation (Roy et al., 2024). The model employs a lattice generation scheme that directly utilizes the first nearest neighbor Warren-Cowley SRO parameter $\tilde Q(x)$7, and quantifies the effects of SRO on the first nearest neighbor three-dimensional site percolation threshold using the large cell Monte Carlo renormalization group method (Roy et al., 2024).

The core result is that the bulk threshold is not fixed by random mixing. Short-range ordering ($\tilde Q(x)$8) raises the threshold, short-range clustering ($\tilde Q(x)$9) lowers it, short-ranged clustering promotes the formation of a dominant spanning cluster, and the scaling exponents of percolation are independent of SRO (Roy et al., 2024). The threshold variation is fitted as

$|x|\to\infty$0

and the thermodynamic link between SRO and pair interaction is

$|x|\to\infty$1

with $|x|\to\infty$2 for FCC (Roy et al., 2024).

The crossover element enters when selective dissolution creates a thin surface layer, so that connectivity must be analyzed between $|x|\to\infty$3-like and $|x|\to\infty$4-like limits. The thickness-dependent crossover relation is

$|x|\to\infty$5

where $|x|\to\infty$6 for $|x|\to\infty$7 and $|x|\to\infty$8 is a non-universal fitting parameter (Roy et al., 2024). As $|x|\to\infty$9 increases, both $2D$00 and the thickness required for percolation decrease for a fixed composition. The model therefore provides a theoretical framework to understand the critical composition of passivating elements for protective oxide formation, and it identifies SRO as a processing parameter for improving corrosion resistance (Roy et al., 2024).

A first-principles extension has been carried out for Cu-Rh alloys by combining a mixed-space cluster expansion trained on density-functional-theory mixing energies, variance-constrained semi-grand canonical Monte Carlo sampling, SRO diagrams, and the FCC SRO-dependent threshold relation (Roy et al., 10 Sep 2025). This produces chemical percolation diagrams over composition and temperature, including

$2D$01

which separate regions where spanning passivating networks do not form, begin to form, or always form (Roy et al., 10 Sep 2025). In that formulation, the percolation crossover model becomes a design map for corrosion-resistant alloys.

6. Methodological caveats, misconceptions, and broader significance

A recurrent misconception is that a reported crossover must always indicate a distinct thermodynamic phase transition. Several studies instead show finite-size, protocol-dependent, or realization-dependent crossovers. In diffusion on percolation lattices, the expected sub-diffusion to classical-diffusion crossover is visible in some individual configurations but is washed out by ensemble averaging (Ghaemi et al., 2010). In quantum percolation with dephasing, the reported metallic regime is interpreted as a crossover regime where the phase-coherence length is comparable to system size, not as a fully established metallic phase in the $2D$02 orthogonal class (Qi et al., 2019). Crossover FSS theory similarly explains anomalous scaling at infinite-system criticality as a mixing effect of standard FSS behaviors around pseudocritical points in event-based ensembles (Li et al., 2024).

A second misconception is that crossover boundaries must be tricritical. The evidence is mixed. The random growth lattice filling model finds continuous percolation transitions for $2D$03, discontinuous transitions for $2D$04, and a tricritical region rather than a sharp tricritical point for $2D$05 (Roy et al., 2017). Equivalent-neighbor $2D$06 percolation, by contrast, shows no evidence of a tricritical point between mean-field and short-range behavior (Ouyang et al., 2018). Long-range $2D$07 percolation reports a pronounced jump in universal quantities at $2D$08 (Liu et al., 22 Sep 2025). These cases indicate that “crossover” covers several distinct scenarios: continuous drift of effective exponents, finite-size rounding, abrupt universality changes at a boundary, and regime mixtures generated by pseudocritical fluctuations.

In hybrid percolation transitions induced by cascading processes, the term also refers to a microscopic two-stage mechanism: a durable critical branching process lasting $2D$09 time is followed by an explosive supercritical process once large loops appear, producing the order-parameter jump (Lee et al., 2016). This usage emphasizes crossover in dynamics rather than in static exponents.

Taken together, the percolation crossover model is not a single formalism but a family of frameworks for describing how percolative behavior changes across scale, geometry, interaction structure, and measurement protocol. Its most mature materials-science realization is the SRO-dependent passivation model for FCC alloys (Roy et al., 2024), but the broader literature shows that the same conceptual apparatus is equally relevant to conductivity, diffusion, epidemic spreading, long-range connectivity, disorder-driven growth, and finite-size scaling (Kim et al., 2014, Santos et al., 2018, Li et al., 2024).