Nonalgebraic first-return probability of a stretched random walk near a convex boundary and its effect on adsorption

The N-step random walk, elongated in the vicinity of a disk (in 2D) or a sphere (in 3D) of radius R, demonstrates a nonalgebraic stretched exponential decay P_{N}∼exp(-constN^{1/3}) for the first-return probability P_{N} in the double-scaling limit N=L/a≫1,R/a≫1 conditioned that L/R=c=const. Stretching means that the length of the walk, L=Na (where a is the unit step length) satisfies the condition L=cR, where c>π and under first return we understand the radial first arrival to a boundary. Both analytic and numerical evidence of the nonalgebraic behavior of P_{N} are provided. Considering the model of a polymer loop stretched (inflated) by external force, we show that nonalgebraic behavior of P_{N} affects the adsorption of a polymer at the boundary of a sticky disk in 2D, manifesting in a first-order localization transition.