Non-algebraic 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 disc (in 2D) or a sphere (in 3D) of radius $R$, demonstrates a non-algebraic stretched exponential decay $P_N\sim \exp (-\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}{N}^{1\mathrm{/}3})$ for the first return probability $P_N$ in the double-scaling limit $N=\frac{L}{a}\gg 1,\frac{R}{a}\gg 1$ conditioned that $\frac{L}{R}=c=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. Stretching means that the length of the walk, $L=Na$ (where $a$ is the unit step length) satisfies the condition $L=cR$, where $c>\pi$ and under “first return” we understand the radial first arrival to a boundary. Both analytic and numerical evidences of the non-algebraic behavior of $P_N$ are provided. Considering the model of a polymer loop stretched (“inflated”) by external force, we show that non-algebraic behavior of $P_N$ affects the adsorption of a polymer at the boundary of a sticky disc in 2D, manifesting in a first order localization transition.