Flips reveal the universal impact of memory on random explorations.

ABSTRACT: Quantifying space exploration is a central question in random walk theory, with direct applications to animal foraging, diffusion-limited reactions, cell motility, and stock markets. The explored domain of memoryless (Markovian) walkers is well understood, but real systems generally exhibit strong memory effects, complicating analysis. We introduce the flip: in one dimension, where the visited territory is [xmin,xmax] , a flip occurs when, after visiting xmax , the walker next discovers xmin-1 rather than xmax+1 (and vice-versa). Although it coincides with the classical splitting probability for Markovian systems, we show that the flip probability reveals the universal impact of memory, or history dependence, on exploration: obeying the universal law π_n ∝ 1/n regardless of the d

SUBMITTER: Bremont J 

PROVIDER: S-EPMC12800135 | biostudies-literature | 2025 Dec

REPOSITORIES: biostudies-literature