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 $P_N\sim exp(-\mathrm{const}{N}^{1/3})$ 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 $\frac{L}{R}=c=\mathrm{const}$. Stretching means that the length of the walk, $L=Na$ (where $a$ is the unit step length) satisfies the condition $L=cR$, where $c>\pi$ 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.

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• Adsorption
• Diffusion
• Phase transitions
• Random walks
• Polymers