Extended Data Fig. 5: Detailed analyses of spectrally modulated propagation in the linear track environment.

a, Copy of Fig. 2d for context and comparative visualization. b, The propagation distributions in panel A are re-expressed as the difference with respect to the baseline (τ = 1, α = 1). c, Each propagation density is plotted against the baseline propagation density (τ = 1, α = 1) on log scales. Note that only superdiffusion results in a non-linear warping of the propagation density. This non-linearity underpins the heavy-tail in the propagation density. d, Small-scale (dashed line) and large-scale (full line) spectral components (that is, generator eigenvectors) are plotted as a function of linear track position. Eigenvalues take a maximum value of 0 and are ordered according to the spatial scale of the corresponding spectral component. For example, eigenvalues close to zero correspond to large-scale spectral components with the spatial scale of the corresponding spectral component decreasing with the eigenvalue which are always less than or equal to zero. Therefore, the scale of a spectral component ϕ_k decreases as the absolute value |λ_k| of its corresponding eigenvalue λ_k increases. e, Spectral power is plotted as a function of eigenvalue on a log scale. As the tempo τ decreases towards zero, the spatiotemporal horizon increases and the propagation density converges on the stationary state distribution. This corresponds to all spectral components with non-zero eigenvalues decaying to zero. Note that diffusion linearly changes the slope of the power spectrum while superdiffusion imposes a non-linear reweighting. This effect underpins the nonlinear time warping in the propagation densities in panel C. f, The characteristic modulation of the power spectrum under diffusion and superdiffusion is demonstrated by the relative power spectrum \(\frac{{s_{\tau ,\alpha }(\lambda )}}{{s_{1,1}(\lambda )}}\) computed as the power ratio with respect to the baseline propagator. Note that the long-range diffusion propagator (α = 1, τ = 0.5, thick red line) downweights small-scale components and upweights large-scale components while superdiffusion (α = 0.5, τ = 1, thick blue line) relatively upweights both small- and large-scale components but suppresses at medium scales. This enables a superdiffusive propagator to jump to remote positions without traversing intermediate locations.