Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917181803986944 |
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| author | Fedotov, Daniil Nechaev, Sergei |
| author_facet | Fedotov, Daniil Nechaev, Sergei |
| contents | 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 $P_N\sim \exp\left(-{\rm const}\, N^{1/3}\right)$ 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 $\frac{L}{R}=c={\rm const}$. 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17829 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption Fedotov, Daniil Nechaev, Sergei Statistical Mechanics Soft Condensed Matter 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 $P_N\sim \exp\left(-{\rm const}\, N^{1/3}\right)$ 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 $\frac{L}{R}=c={\rm const}$. 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 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. |
| title | Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption |
| topic | Statistical Mechanics Soft Condensed Matter |
| url | https://arxiv.org/abs/2506.17829 |