  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.
