# Persistent and anti-Persistent Motion in Bounded and Unbounded Space: Resolution of the First-Passage Problem

New Journal of Physics（2024）

Abstract

The presence of temporal correlations in random movement trajectories is a
widespread phenomenon across biological, chemical and physical systems. The
ubiquity of persistent and anti-persistent motion in many natural and synthetic
systems has led to a large literature on the modelling of temporally correlated
movement paths. Despite the substantial body of work, little progress has been
made to determine the dynamical properties of various transport related
quantities, including the first-passage or first-hitting probability to one or
multiple absorbing targets when space is bounded. To bridge this knowledge gap
we generalise the renewal theory of first-passage and splitting probabilities
to correlated discrete variables. We do so in arbitrary dimensions on a lattice
for the so-called correlated or persistent random walk, the one step
non-Markovian extension of the simple lattice random walk in bounded and
unbounded space. We focus on bounded domains and consider both persistent and
anti-persistent motion in hypercubic lattices as well as the hexagonal lattice.
The discrete formalism allows us to extend the notion of the first-passage to
that of the directional first-passage, whereby the walker must reach the target
from a prescribed direction for a hitting event to occur. As an application to
spatio-temporal observations of correlated moving cells that may be either
repelled or attracted to hard surfaces, we compare the first-passage statistics
to a target within a reflecting domain depending on whether an interaction with
the reflective interface invokes a reversal of the movement direction or not.
With strong persistence we observe multi-modality in the first-passage
distribution in the former case, which instead is greatly suppressed in the
latter.

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Key words

first-passage probability,correlated random walk,lattice random walk,occupation probability,splitting probability

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