Phase transitions in persistent and run-and-tumble walks

Abstract

We calculate the large deviation function of the end-to-end distance and the corre-spondingextension-versus-forcerelationfor(isotropic)randomwalks,onandoff-lattice,withandwithoutpersistence,andinanyspatialdimension.Foroff-latticerandomwalkswith persistence, the large deviation function undergoes a first order phase transitionin dimensiond>5. In the corresponding force-versus-extension relation, the extensionbecomes independent of the force beyond a critical value. The transition is anticipatedin dimensionsd=4 andd=5, where full extension is reached at a finite value of theapplied stretching force. Full analytic details are revealed in the run-and-tumble limit.Finally,on-latticerandomwalkswithpersistencedisplayasofteningphaseindimensiond=3 and above, preceding the usual stiffening appearing beyond a critical value of theforce.