Directed Percolation (DP) Universality Class

• Directed Percolation (DP) Universality Class defines absorbing-state transitions with characteristic power-law scaling and critical exponents in systems lacking spontaneous activity.
• For ε = 0, the model exhibits a clear phase transition with a critical threshold (p_c ≈ 0.6320) and well-measured scaling laws, confirming DP-class behavior.
• Introducing spontaneous activity shifts the system to a quasi-critical regime marked by finite susceptibility peaks and distinct thresholds for optimal response and correlation.

The three-species dynamical model of semi-directed percolation explores an absorbing-state phase transition and its modification under spontaneous activity input. This model demonstrates both critical and quasi-critical regimes, elucidating the relationship between intrinsic system dynamics and external or autonomous excitation. Rigorous numerical evidence establishes that in the absence of spontaneous activity, the model realizes the directed percolation (DP) universality class. When spontaneous activity is present, the critical point is destroyed; however, quasi-critical behavior with nontrivial signatures persists.

1. Model Description and Dynamics

The model consists of three species represented by discrete site states (e.g., 0, 1, 2), with dynamical rules that naturally generate semi-directed percolation clusters. For ε = 0 (no spontaneous activity), the system allows only activity initiated by existing active regions, producing a strict absorbing state (all sites in state 0). For ε > 0, spontaneous activity can arise anywhere, obliterating true absorption and inducing steady but fluctuating activity everywhere.

Monte Carlo simulations are performed on large one-dimensional periodic lattices, tracking time evolution under both a fully active initial condition and a single active seed. At each time step, the rules governing state changes introduce local propagation, recovery to a susceptible state, and (if ε > 0) spontaneous activation.

2. Critical Threshold, Exponents, and Scaling Relations

For ε = 0, the system displays a classic absorbing-state phase transition at a threshold value p_c ≈ 0.6320(2). The measured exponents from large-scale simulations correspond to those of (1+1)-dimensional DP:

Critical Quantity	Scaling Law	DP Value (Measured)
Order parameter	ρₛ ∝ (p − p_c)^β	β ≈ 0.276(1)
Survival probability	pₛ(t) ∼ t^–δ at p = p_c	δ ≈ 0.16(1)
Active number growth	N(t) ∼ t^θ	θ ≈ 0.31(2)
Spatial correlation	g₍⊥₎(r, t) ∼ r^–β/ν₍⊥₎	β/ν₍⊥₎ ≈ 0.252
Temporal correlation	g₍∥₎(Δt) ∼ (Δt)^–β/ν₍∥₎	β/ν₍∥₎ ≈ 0.159

Finite-size scaling forms confirm asymptotic DP-class behavior:

• ρ(t) = t^–β/ν₍∥₎ f₁(t/L^z)
• ρₛ(p, L) = L^–β/ν₍⊥₎ f₃(L^1/ν₍⊥₎(p − p_c))

3. Non-equilibrium Phase Transition

Absorbing state phase transitions separate regimes with sustained activity from those where the system falls into an absorbing, inactive state. For ε = 0, the active phase shows persistent, spreading clusters, while the absorbing phase is inaccessible to reactivation. At criticality, the order parameter and survival probability exhibit power-law decay, with exponents precisely matching the DP universality class, including the equality β = β' for the density and survival exponents.

The time dependence of order parameter decay and cluster growth from a single seed reflects DP scaling laws—the temporal and spatial correlations are consistent with scale-free propagation at threshold.

4. Spontaneous Activity, Quasi-Criticality, and the Widom Line

Introducing spontaneous activity (ε > 0) eliminates the strict absorbing state—activity will always spontaneously reappear, so the system does not undergo a true phase transition. Instead, quasi-critical behavior emerges:

• The dynamic susceptibility, χ = L [〈ρ(t)²〉 − 〈ρ(t)〉²], which diverges at a true critical point, now shows a finite, pronounced peak as a function of the control parameter p.
• This peak defines a non-equilibrium Widom line, a locus of maximum response (fluctuations) but not true divergence.
• The location of this susceptibility maximum, p*₍χ₎, shifts and broadens with increasing ε.

A key observation is that long-range spatial and temporal correlations with power-law decay persist, but the parameter value where these correlations are maximized (p*₍corr₎) differs from p*₍χ₎ when ε > 0. For small ε these points are close; as ε increases the difference becomes pronounced, revealing that the system supports two effective thresholds separating maximal response from maximal scale-free correlations.

5. Spatial and Temporal Correlations

For ε = 0 just above p_c, the equal-time spatial correlation function g₍⊥₎(r, t) and the temporal autocorrelation function g₍∥₎(Δt) both exhibit power-law decays, providing direct access to the DP exponents β/ν₍⊥₎ and β/ν₍∥₎, respectively. With spontaneous activity (ε > 0), these power laws remain observable but only at a control parameter slightly offset from the susceptibility maximum, consistent with the dual-threshold scenario.

6. Implications and Scope

The demonstration that the DP universality class remains robust for the three-species semi-directed percolation model in the absence of spontaneous activity confirms the broad applicability of DP scaling to absorbing-state transitions. The emergence of quasi-critical behavior under persistent external drive or spontaneous activity, characterized by a non-equilibrium Widom line and bifurcated thresholds for maximum response and correlation, provides a framework for understanding similar phenomena in biological networks, particularly neuronal dynamics where spontaneous firing is always present.

These results suggest that in realistic systems with persistent driving or background noise, true phase transitions may be replaced by sharp crossover regions, yet critical-like signatures—power-law correlations and peaks in susceptibility—can serve as practical markers of quasi-criticality.

7. Mathematical Summary

Key formulas governing the critical and quasi-critical behavior include:

• Order parameter scaling: ρₛ ∝ (p − p₍c₎)^β
• Survival probability: Pₛ ∝ (p − p₍c₎)^β'
• Decay at criticality: ρ(t) ∼ t^–α, pₛ(t) ∼ t^–δ
• Active cluster growth: N(t) ∼ t^θ
• Correlations: g₍⊥₎(r, t) ∼ r^–β/ν₍⊥₎, g₍∥₎(Δt) ∼ (Δt)^–β/ν₍∥₎
• Dynamic susceptibility (response): χ = L (〈ρ(t)²〉 − 〈ρ(t)〉²)
• Finite-size scaling: ρ(t) = t^–β/ν₍∥₎ f₁(t/L^z), ρₛ(p, L) = L^–β/ν₍⊥₎ f₃(L^1/ν₍⊥₎(p − p₍c₎))

Summary Table: Critical and Quasi-Critical Regimes

Regime	ε	Absorbing Phase Exists	Susceptibility Divergence	Spatial/Temporal Power Laws	Distinct “Thresholds”
Critical	0	Yes	Yes	At p = p₍c₎	No
Quasi-Critical	>0	No	Finite peak at p*₍χ₎	At p*₍corr₎ ≠ p*₍χ₎	Yes

This framework establishes that the three-species semi-directed percolation model realizes the DP universality class in the absence of spontaneous activity, while for nonzero spontaneous activation, it exhibits quasi-critical phenomena. The identification of two distinct operational thresholds for susceptibility and correlations underlines important structural and dynamical nuances relevant to a wider class of driven, nonequilibrium systems (Jasna et al., 29 Aug 2025).