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Emergent heavy-tailed distributions from a Markovian random walk

Henrique S. Lima* and Evaldo M. F. Curado

  • *Contact author: hslima94@cbpf.br
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Phys. Rev. E 114, 034114 – Published 9 September, 2026

DOI: https://doi.org/10.1103/f1bd-1242

Abstract

The emergence of heavy-tailed statistics in complex systems is conventionally attributed to nonlocal stochastic jumps or non-Markovian memory. Here, we present a one-dimensional random walk where power-law behaviors arise instead from a strictly local, discrete-time Markovian mechanism. The step length is governed by a deterministic function of the walker's position, establishing a positive feedback loop that induces strong effective correlations along the trajectories. Through analytical derivations in the continuum limit and extensive numerical simulations, we show that this rule yields a robust, non-Gaussian stationary state. The exact analytical solution is obtained in the closed form of a symmetric, Lorentz-like distribution ρst(x)∝(|x|/l+rΔx)−2, confirming asymptotic power-law tails that decay as |x|−2 over six decades. Furthermore, by employing the Onsager-Machlup path-integral formalism, we demonstrate that effective velocity and acceleration acquire physical meaning along the shortest fluctuation trajectories. Crucially, we find that a nonzero initial acceleration acts as the fundamental mechanism driving the walker away from the origin, ensuring both the emergence of scale-free statistics and the normalizability of the stationary distribution. This minimal pathway provides a microscopic foundation for the widespread −2 power law observed across multidisciplinary complex systems.

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Phys. Rev. E 114, 034114– Published 9 September, 2026
  • Received 21 May 2026
  • Accepted 25 August 2026
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