Semi-Directed Percolation Cluster

• Semi-directed percolation clusters are structures that interpolate between isotropic and directed percolation, featuring anisotropic connectivity and crossover phenomena.
• Models such as the biased directed percolation (BDP) use distinct occupation probabilities to capture directional biases and critical scaling behavior.
• These clusters reveal universal properties through scaling exponents and crossover laws, informing studies on network dynamics and phase transitions.

A semi-directed percolation cluster is a percolation structure that interpolates between isotropic (undirected) and fully directed percolation by allowing propagation or connectivity along different spatial directions with distinct probabilities or biases. These clusters emerge in models where directionality is neither fully imposed nor fully absent, often leading to a spectrum of geometric and critical phenomena sensitive to the degree of anisotropy or asymmetry in connectivity. The notion of a semi-directed percolation cluster appears in both theoretical lattice models and dynamical systems exhibiting absorbing-state phase transitions, and is characterized by crossover phenomena, scaling exponents, and universality classes that can differ from those of standard isotropic or directed percolation.

1. Formal Definition and Model Construction

A central mathematical realization of the semi-directed percolation cluster is provided by the biased directed percolation (BDP) model (Zhou et al., 2011). In this model, each bond (edge) between neighboring sites on a lattice is independently in one of three states: occupied with a downstream orientation, occupied with an upstream orientation, or unoccupied. The occupation probabilities are defined as

$$p_{\downarrow} = p \cdot p_d, \qquad p_{\uparrow} = p \cdot (1 - p_d)$$

where $p$ is the average (total) bond occupation probability and %%%%1%%%% is the parameter that controls the net directional bias. When $p_d = 1/2$, the model is isotropic percolation (IP); when $p_d = 0$ or $p_d = 1$, the process is fully anisotropic, recovering the standard directed percolation (DP) universality class. For intermediate $p_d$, the model is "semi-directed," admitting a variable degree of directionality.

Beyond the BDP model, generalized semi-directed percolation clusters are also defined on more abstract graphs. For instance, on directed $d$-regular vertex-transitive graphs (Tanaka, 2016), each vertex randomly selects a subset of neighbors to connect, and the directionality and local connectivity type are determined by a probability vector on subset sizes. Weak and strong cluster connectivity definitions are used: weak connectivity allows for paths regardless of orientation, while strong connectivity requires mutual (bidirectional) linkage. The Lightning Model on $\mathbb{Z}^2$ (Campbell et al., 2021) generates a random directed subgraph using i.i.d. vertex potentials and an edge-opening criterion with a tunable tolerance parameter $\epsilon$.

The essential feature across these constructions is that connectivity is neither fully symmetric nor strictly one-way, enabling exploration of universal behaviors as systems interpolate between traditional percolation classes.

2. Cluster Structure, Connectivity, and Scaling Regimes

The cluster geometry in semi-directed percolation models is sensitive to the degree and nature of anisotropy. In the BDP model (Zhou et al., 2011), clusters are constructed using the Leath-Alexandrowicz (breadth-first growth) method from a seed site, respecting directional occupation probabilities. Observables such as the number of activated sites $N(s)$ at chemical shell $s$, the survival probability $P(s)$, and the modified gyration radius $R(s)$ satisfy scaling laws: $$A(s, \epsilon) \sim s^{Y_A} \mathcal{A}( \epsilon s^{Y_\epsilon} ),$$ with $\epsilon = p - p_c$, critical threshold $p_c$, and $Y$ exponents specific to the chosen observable. For isotropic percolation ($p_d = 1/2$), the shortest-path (chemical distance) exponent, $d_\mathrm{min}$, governs longitudinal scaling, while in the semi-directed and DP limits, exponents such as survival decay ($\delta$), spreading ($\eta$), and longitudinal correlation length ($\nu_\parallel$) become relevant.

Anisotropic percolation systems with frozen disorder or columnar defects (Grassberger et al., 2017) exhibit strongly anisotropic clusters: the longitudinal (preferred direction) extent $R_z(t)$ grows ballistically ($R_z \sim t$), while the transverse spread $R_{xy}(t) \sim t^{1/2}$ is diffusive. This anisotropy is also reflected in correlation lengths and scaling exponents, and below threshold such systems may exhibit a Griffiths phase characterized by continuously varying non-universal exponents.

In general, semi-directed percolation clusters are expected to exhibit scaling behavior that interpolates between isotropic and fully directed fixed points, with crossovers dictated by the degree of bias.

3. Phase Transitions, Critical Exponents, and Universality

The transition from non-percolating to percolating phases in semi-directed models retains many haLLMarks of second-order percolation transitions, including the emergence of an infinite or "spanning" cluster. In the BDP model (Zhou et al., 2011), Monte Carlo simulations reveal that the critical exponents governing cluster growth, survival probabilities, and correlation lengths remain in the DP universality class for any $p_d \ne 1/2$. The isotropic percolation point ($p_d = 1/2$) forms a distinct fixed point with its own critical exponents, such as the chemical dimension $d_\mathrm{min}$.

The renormalization-group flow is controlled by an asymmetric (relevant) scaling field proportional to $p_d - 1/2$. The crossover exponent $\phi$ is given by the ratio of the scaling field exponent $Y_{\epsilon_d}$ to $Y_\epsilon = 1/\nu_\parallel$. Crossover scaling is observed: $1 - p_c \propto |p_d - 1/2|^{1/\phi}$ Numerical values of the exponents extracted from finite-size scaling (e.g., $Y_{\epsilon_d} \simeq 0.500(5)$ in 2D (Zhou et al., 2011)) confirm the renormalization predictions.

In dynamical models, such as a three-species system generating semi-directed clusters (Jasna et al., 29 Aug 2025), the absorbing-state phase transition falls into the DP class, with steady-state density and survival probability exponents (e.g., $\alpha \simeq 0.16$ for the order parameter decay, $\beta \simeq 0.276$ for the steady-state scaling).

A key result is that any anisotropy or directional bias is a relevant perturbation at the isotropic fixed point, driving the system into the directed percolation universality class. Only at the precisely isotropic point does a different universality class appear.

4. Crossover Phenomena and Scaling Field Relevance

The existence of a relevant asymmetric field in the BDP model causes rapid crossover from isotropic to directed percolation scaling. For any small but finite $p_d - 1/2$, observable quantities rapidly acquire DP scaling properties, and the isotropic scaling is unstable under anisotropic perturbations (Zhou et al., 2011). The renormalization flow can be visualized in the $(p, p_d)$ parameter space, with scaling functions for observables exhibiting universal forms once expressed in terms of appropriate scaled variables.

Crossover scaling laws and dimensionless ratios (such as $Q(s)$, the shell-wise ratio of active sites) are used to identify critical points and quantify the rate of crossover (e.g., through data collapse using $Q(s)$ vs. $(p_d - 1/2)^2$ for fixed $p$ near $p_c$). The relevance of the asymmetric field is confirmed by the observed consistency with scaling predictions.

In models including additional types of disorder, such as columnar removal (Grassberger et al., 2017), Griffiths phases can emerge below the critical point, with non-universal, power-law tails in cluster size or survival, determined by rare-region effects associated with frozen disorder aligned with the preferred direction.

5. Applications and Extensions in Network and Statistical Models

The theoretical framework of semi-directed percolation clusters has implications for a range of systems:

• Stochastic processes and random walks: The backbone of a semi-directed percolation cluster serves as the configuration space for directed random walks, which exhibit diffusive scaling, law of large numbers, and central limit theorem convergence both in annealed and quenched settings. Coalescing random walks on such backbones converge to the Brownian web after diffusive rescaling, with large-scale universality matching that of the full lattice but with modified variance due to the underlying cluster geometry (Birkner et al., 2012, Birkner et al., 2018).
• Directed epidemics and message passing: Artificial and real-world networks with mixed directed and undirected connectivities are amenable to message-passing and generating-function analyses (Widder et al., 2021). In such networks, semi-directed clusters describe the giant component structure and thresholds for epidemic outbreaks, with percolation transitions determined by the spectral radius of appropriate non-backtracking (Hashimoto) matrices (Hamilton et al., 2015).
• Cluster aggregation and hybrid transitions: In dynamics where link formation follows semi-directed (for example, preferential merging or directionally biased rules), hybrid percolation transitions may arise with a mixture of discontinuous (first-order) and critical (second-order) behavior and continuously varying critical exponents (Cho et al., 2015).
• Dynamical phase transitions and spontaneous activity: Systems with spontaneous activation (e.g., models of neuronal avalanches) abolish strict absorbing-state transitions but preserve quasi-critical phenomena. Distinct effective thresholds appear for susceptibility maxima and scale-free correlations, a phenomenon also observed in semi-directed percolation-type dynamics (Jasna et al., 29 Aug 2025).

6. Mathematical Formulations and Observables

Key observables in semi-directed percolation cluster analysis include:

• Order parameter (density of active sites/size of infinite cluster): $\rho$
• Survival probability: $P_s(t) \sim t^{-\delta}$ at criticality
• Number of activated sites at chemical shell $s$: $N(s)$
• Gyration radius and shortest-path scaling: $d_{\mathrm{min}}$ (chemical dimension)
• Dimensionless ratios for finite-size scaling: $Q(s) = (2s)/(s)$
• Critical exponents: $\eta$, $\delta$, $\nu_{\parallel}$, and $\nu_{\perp}$ (for longitudinal and transverse correlation lengths)
• Scaling laws:

  $$A(s, \epsilon) \sim s^{Y_A} \mathcal{A}(\epsilon s^{Y_\epsilon}), \qquad (\epsilon = p - p_c)$$

  and crossover exponents characterized through

  $1 - p_c \propto |p_d - 1/2|^{1/\phi}$

• Response and susceptibility: $$\chi = L \left[ \langle \rho(t)^2 \rangle - \langle \rho(t) \rangle^2 \right]$$

These observables are accessible through both numerical (Monte Carlo) simulation and analytic techniques (generating functions, finite-size scaling, renormalization group).

7. Implications, Applications, and Universality

Semi-directed percolation clusters provide a unified conceptual and computational framework for understanding the interplay of symmetry, directionality, and criticality in random media. Models incorporating both isotropic and anisotropic connectivity, random walks in dynamic random environments, network robustness, and dynamical processes with absorbing states are all amenable to analysis via this framework.

The key findings across model realizations are:

• The universality class is controlled by the presence or absence of strict symmetry; any directional bias is a relevant perturbation that generically yields DP-class scaling.
• Crossover phenomena manifest rapidly near the isotropic point, with clear scaling predictions for observables and exponents.
• Quasi-critical regimes can emerge under processes such as spontaneous activation, with effective scaling regimes and Widom lines replacing strict phase transitions.
• The structural and critical properties of semi-directed percolation clusters directly inform the understanding of percolative transitions in both equilibrium and non-equilibrium settings, ranging from materials science to population dynamics and neural activity.

Semi-directed percolation cluster theory thus bridges the gap between the classical paradigms of undirected and directed percolation, making it a central object of paper for the analysis of anisotropic critical phenomena and complex network structures.