Emergent heavy-tailed distributions from a Markovian random walk

Abstract

The emergence of heavy-tailed statistics in complex systems is conventionally attributed to non-local 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, $\rho_{\text{st}}(x) \propto (|x|/l+r\Delta x)^{-2}$, 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 non-zero 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 new microscopic foundation for the widespread $-2$ power law observed across multidisciplinary complex systems.