Deterministic scale-invariant dynamics in a logistic Game-of-Life model

TL;DR

The paper demonstrates that a purely deterministic extension of Conway's Game of Life, the logistic GOL, can exhibit scale-invariant dynamics without external noise. By expanding the local state space to a Cantor-like set and tuning the update strength via $\lambda$, the authors identify three asymptotic phases separated by two deterministic critical points $\lambda_A$ and $\lambda_P$. At $\lambda_A$, the system undergoes a static-dynamic transition with SOC-like, power-law distributions of zero-state clusters, while at $\lambda_P$ a deterministic percolation transition emerges with a fractal cluster geometry and an unconventional Fisher exponent $\tau\approx1.81$; wrapping-probability analyses locate the thermodynamic threshold at $\lambda_P(N\to\infty)\approx0.86134$. Across these transitions, the cluster-size distributions, capacity and fractal dimensions, and KS-based power-law tests reveal deterministic routes to scale invariance, distinct from traditional stochastic SOC frameworks. Importantly, the asymptotic statistics are shown to be independent of initial randomness, highlighting a robust, purely deterministic mechanism for criticality with potential implications for no-noise realizations of SOC and percolation phenomena. The results broaden the landscape of critical phenomena by showing that purely deterministic local rules can yield both percolation-like and SOC-like scale invariance, with unconventional exponents and self-organized dynamics arising without external perturbations.

Abstract

Scale invariance is a hallmark of criticality in complex dynamical systems. While random external inputs or tunable stochastic interactions are typically required to produce critical behavior, it remains unclear whether scale-invariant dynamics can emerge from purely deterministic interactions. Here, we address this question by studying the asymptotic dynamics of the logistic Game of Life (GOL), a deterministic-parameter extension of Conway's GOL. In this system, we identify three distinct asymptotic phases separated by two fundamentally different critical points. The first critical point, associated with an unusual form of self-organized criticality, separates a sparse-static phase from a sparse-dynamic phase. The second critical point corresponds to a deterministic percolation transition between the sparse-dynamic phase and a third, dense-dynamic phase. In addition, we observe power-law cluster size distributions with unconventional critical exponents not found in standard equilibrium systems. Overall, our work paves the way for studying emergent scale invariance in purely deterministic systems.

Figures (19)

• Figure 1: The logistic Game of Life.(a) Summary of the update rules of Conway’s Game of Life (middle) and the logistic Game of Life (right). In the logistic Game of Life, each lattice site carries a continuous state $s\in[0,1]$, and the control parameter $\lambda$ sets the update strength. For a site at time $t$ with state $s^t$, we compute the Moore sum $m^t$ and define the increment $\Delta s \equiv s^{t+1}-s^t$, which depends on three thresholds $t_1<t_2<t_3$ that partition the neighborhood-sum axis into stability, growth, and decay regimes. Conway’s Game of Life is recovered in the discrete limit ($s\in\{0,1\}$) at $\lambda=1$ with the usual birth/survival rules. (b) Illustrative snapshots of asymptotic configurations of the logistic Game of Life at representative values of the control parameter $\lambda$. The colorbar encodes the local site state $s$: colors toward dark purple indicate $s\simeq 0$, whereas colors toward bright yellow indicate $s\simeq 1$.
• Figure 2: Three distinct asymptotic phases in the logistic GOL separated by two critical points.(a) Asymptotic average activity $\langle{A}\rangle$ (solid blue) and the size of the largest cluster $\langle{\mathcal{S}_1}\rangle/N^2$ (solid red) computed against $\lambda$. The data indicate two critical points: (i) $\lambda_{\mathrm{A}}=0.875$ (blue dashed line), the boundary between a sparse-static (I) and sparse-dynamic (II) asymptotic phase; (ii) $\lambda_{\mathrm{P}}=0.86055$ (red dashed line), where fragmentation of the largest cluster defines the boundary between phase II and dense-dynamic (III) phase. (b) The susceptibility of activity $\langle{\chi}\rangle$, plotted against $\lambda$, reaches its maximum at $\lambda_{\mathrm{A}}$.
• Figure 3: Deterministic cluster dynamics reveals a percolation transition in the logistic GOL.(a) Top panels display snapshots of the asymptotic states of the logistic GOL at distinct $\lambda$ values in the range $[0.850,0.875]$ Bottom panels show the corresponding five largest clusters masked in different colors (ranking in panel b), while the red dashed line marks $\lambda_{\mathrm{P}}$. (b) Sizes of the largest clusters $\langle{\mathcal{S}_2}\rangle \sim \langle{\mathcal{S}_5}\rangle$ plotted against $\lambda$, where the index $i$ indicates the size rank of the cluster. The curves differ only by scaling when $\lambda<\lambda_{\mathrm{P}}$. The inset displays the logarithmic evolution of cluster sizes, with the largest zero-state cluster (dark red) percolating as $\lambda$ increases. The evolution of (c) capacity dimensions $d_{\mathrm{c}}$ of the largest clusters and their (d) corresponding standard deviations $\sigma_c$ computed as functions of $\lambda$. (e) Scaling behavior of the largest cluster with lattice size ($N$) around $\lambda_{\mathrm{P}}$. In the 'subcritical' regime ($\lambda<\lambda_{\mathrm{P}}$, left), cluster sizes $\langle \mathcal{S}_i(N) \rangle$ follow a logarithmic trend. Around the critical point ($\lambda = \lambda_{\mathrm{P}}$, middle), the clusters scale as power laws, where the exponent of the largest cluster defines the fractal dimension $d_{\mathrm{f}}$. In the 'supercritical' regime ($\lambda > \lambda_{\mathrm{P}}$, right), the largest cluster $\langle \mathcal{S}_1 \rangle$ scales with the system's dimension, spanning the lattice. The dashed lines show the corresponding fits to the collected means; error bars denote the standard errors in the estimated means arising from fluctuations in $\langle \mathcal{S}_i\rangle$.
• Figure 4: Wrapping probabilities around the percolation transition.(a) Wrapping probability in either the horizontal or vertical direction, $R_{\mathrm{W}}^{(\mathrm{e})}$. (b) Wrapping probability in both directions simultaneously, $R_{\mathrm{W}}^{(\mathrm{b})}$. (c) Wrapping probabilities in the horizontal and vertical directions, $R_{\mathrm{W}}^{(\mathrm{h})}$ and $R_{\mathrm{W}}^{(\mathrm{v})}$. The first column shows the evolution of the corresponding wrapping probabilities as a function of $\lambda$, while the second column shows their evolution as a function of system size $N$ for selected $\lambda$ values chosen around the thermodynamic percolation point $\lambda_{\mathrm{P}}(N \to \infty)$. The curves for different system sizes $N$ intersect at a common fixed point $\lambda_P(N \to \infty) \simeq 0.86134$ (red dashed line), with corresponding wrapping probabilities marked at $R_{\mathrm{W}}^{(\mathrm{e})} \approx 0.7163, R_{\mathrm{W}}^{(\mathrm{b})} \approx 0.3667, R_{\mathrm{W}}^{(\mathrm{h})} = R_{\mathrm{W}}^{(\mathrm{v})} \approx 0.5415$ (black dashed lines).
• Figure 5: Behavior of cluster size distribution around $\lambda_{\mathrm{P}} = 0.86055$. The empirical complementary cCDFs with logarithmic-binning are shown in blue, with the fitted power-law in orange, for $\lambda$ values (a) below, (b) close, and (c-d) above $\lambda_{\mathrm{P}}$. The x-axis starts from the optimal $\mathcal{S}_\mathrm{min}$ determined by the KS method. The cluster size distribution becomes a power law (with exponential cutoff) only very close to the critical point $\lambda_{\mathrm{P}}$.