The Topology of Rayleigh-Levy Flights in Two Dimensions

TL;DR

The paper extends Rayleigh-Lévy flights to two dimensions to study non-Gaussian, long-range correlated cosmic-density-like fields using one-point statistics and Minkowski functionals. It derives mean-field results for the density CGF and PDF and develops beyond-mean-field corrections for the density PDF and iso-density perimeter $W_1$, validating them against extensive simulations. The results show that BMF predictions accurately reproduce $P(\rho)$ and $W_1$ across a wide variance range, while MF remains reliable in high-density tails, with finite-volume effects carefully accounted for. The work strengthens Rayleigh-Lévy flights as a versatile analytic toy model for nonlinear structure formation and provides a pathway to study higher-order topology and critical-point statistics in a non-Gaussian setting.

Abstract

Rayleigh-Lévy flights are simplified cosmological tools which capture certain essential statistical properties of the cosmic density field, including hierarchical structures in higher-order correlations, making them a valuable reference for studying the highly non-linear regime of structure formation. Unlike standard Markovian processes, they exhibit long-range correlations at all orders. Following on recent work on one dimensional flights, this study explores the one-point statistics and Minkowski functionals (density PDF, perimeter, Euler characteristic) of Rayleigh-Lévy flights in two dimensions. We derive the Euler characteristic in the mean field approximation and the density PDF and iso-field perimeter $W_{1}$ in beyond mean field calculations, and validate the results against simulations. The match is excellent throughout, even for fields with large variances, in particular when finite volume effects in the simulations are taken into account and when the calculation is extended beyond the mean field.

Figures (7)

• Figure 1: Two-dimensional Rayleigh-Lévy flight with parameters $\alpha = 3/2$, $\xi_0\sim 2.47$ ($\ell_{0} = 0.01$), density $N_{\rm pts} = 40$ points per pixel, box width $L = 4000 \times \Delta$, resolution $\Delta = 0.25$, and Gaussian smoothing scale $R = 1$.
• Figure 2: Top panel: measured correlation function for Lévy flights with $\alpha = 1.5$, $\ell_{0} = 0.02$ (red points/error bars). The blue dashed line is the expected large $r \gg \ell_{0}$ behaviour $\zeta(r) \sim r^{\alpha-D}$, but the actual dependence is slightly steeper, with an effective measured power law $\alpha \simeq 1.43$ (solid blue line). Bottom panel: the measured Skewness ${\cal S}_3$ from Lévy flights with different $\xi_{0}$ variances (red points and error bars) with $\alpha = 3/2$. The error bars correspond to the error on the mean from $N_{\rm real} = 100$ realisations each. The beyond mean field prediction yields a better fit to the measured skewness compared to the mean field (black dashed line), but there is a slight ambiguity in recovering the exact BMF prediction due to the $\alpha$ dependence of the statistic (cf. blue solid/dashed lines and shaded region).
• Figure 3: Left: Two-dimensional Lévy flight probability distribution function and right: difference with respect to 'mean field' (MF) with parameters $\alpha=3/2$, $\xi_0=0.08, 0.88, 2.47$ ( top-bottom panels), $L = 4000\times \Delta$ (box width), $N_{\rm pts} = 40 \times 2000^2$ (number of steps), $\Delta = 0.25$ (resolution), and $R = 1$ (smoothing scale). The error bars show the standard deviation in the left panels and the error on the mean for $100$ realisations in the right panels. The points the sample means. The corresponding standard deviations of the fields are given by $\sigma_{0} = \sqrt{\xi_{0}} = 0.28, 0.94, 1.57$ ( top-bottom). The BMF prediction agrees well with the simulations even in the non-perturbative, non-Gaussian regime.
• Figure 4: Left: excursion set perimeter length $W_{1}$ and right Euler characteristic -- measured (red points/error bars), mean-field expectation value (black dashed lines) and beyond mean field with $\alpha =1.5$ (blue dashed lines) and $\alpha = 1.43$ (blue solid lines). The error bars represent the standard deviation, and the points the sample means. Top to bottom panels are the cases $\xi_0=0.08, 0.88,2.47$, respectively, with other parameters $\Delta = 0.25$, $R = 1$, $L=4000 \times \Delta$, $\alpha = 3/2$, $N_{\rm pts} = 40 \times 2000^{2}$. Once again the agreement is excellent, especially at the high density end. Note that the beyond mean field prediction for $\chi$ (right panels) is beyond the scope of this paper.
• Figure 5: Summary statistics $P(\rho)$ ( top left), $W_{1}$ ( bottom left) and $\chi$ ( bottom right) for a set of $N_{\rm real} = 100$ realisations of a highly non-Gaussian field with $\xi_{0} =27$ and other parameters $\Delta = 0.25$, $R = 1$, $L=4000 \times \Delta$, $\alpha = 3/2$, $N_{\rm pts} = 40 \times 2000^{2}$. The difference between $P(\rho)$ and the mean field prediction is also presented ( top right). Error bars show the standard deviation, and the points the sample means. The BMF estimates of the PDF (blue solid lines, top panels) and $W_{1}$ ( bottom left panel) remain an excellent approximation for $\xi_{0} > 20$. The mean field limit is a good approximation for all statistics in the high density tails.