Phase transitions in persistent and run-and-tumble walks

TL;DR

This paper analyzes phase transitions in various types of random walks, revealing first-order and softening phase transitions in high-dimensional and on-lattice models, with detailed analytic results especially in the run-and-tumble limit.

Contribution

It provides a comprehensive calculation of large deviation functions and force-extension relations for persistent and run-and-tumble random walks across different dimensions and lattice types, revealing new phase transition phenomena.

Findings

First-order phase transition in off-lattice persistent walks for d>5.

Force independence of extension beyond critical force.

Softening phase in on-lattice persistent walks in d≥3.

Abstract

We calculate the large deviation function of the end-to-end distance and the corresponding extension-versus-force relation for (isotropic) random walks, on and off-lattice, with and without persistence, and in any spatial dimension. For off-lattice random walks with persistence, the large deviation function undergoes a first order phase transition in dimension $d> 5$. In the corresponding force-versus-extension relation, the extension becomes independent of the force beyond a critical value. The transition is anticipated in dimensions $d=4$ and $d=5$, where full extension is reached at a finite value of the applied stretching force. Full analytic details are revealed in the run-and-tumble limit. Finally, on-lattice random walks with persistence display a softening phase in dimension $d=3$ and above, preceding the usual stiffening appearing beyond a critical value of the force.