Scaling limit of a one-dimensional polymer in a repulsive i.i.d. environment
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916180861648896 |
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| author | Bouchot, Nicolas |
| author_facet | Bouchot, Nicolas |
| contents | The purpose of this paper is to study a one-dimensional polymer penalized by its range and placed in a random environment $ω$. The law of the simple symmetric random walk up to time $n$ is modified by the exponential of the sum of $βω_z - h$ sitting on its range, with~$h$ and $β$ positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size $c_h n^{1/3}$ with $c_h = π^{2/3} h^{-1/3}$. Here we study how disorder influences finer quantities. If the random variables $ω_z$ are i.i.d.\ with a finite second moment, we prove that the left-most point of the range is located near $-u_* n^{1/3}$, where $u_* \in [0,c_h]$ is a constant that only depends on the disorder. This contrast with the homogeneous model (i.e. when $β=0$), where the left-most point has a random location between $-c_h n^{1/3}$ and $0$. With an additional moment assumption, we are able to show that the left-most point of the range is at distance $\mathcal U n^{2/9}$ from $-u_* n^{1/3}$ and the right-most point at distance $\mathcal V n^{2/9}$ from $(c_h-u_*) n^{1/3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$ are constants that depend only on $ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scaling limit of a one-dimensional polymer in a repulsive i.i.d. environment Bouchot, Nicolas Probability 82B44, 60G50, 60G51 The purpose of this paper is to study a one-dimensional polymer penalized by its range and placed in a random environment $ω$. The law of the simple symmetric random walk up to time $n$ is modified by the exponential of the sum of $βω_z - h$ sitting on its range, with~$h$ and $β$ positive parameters. It is known that, at first order, the polymer folds itself to a segment of optimal size $c_h n^{1/3}$ with $c_h = π^{2/3} h^{-1/3}$. Here we study how disorder influences finer quantities. If the random variables $ω_z$ are i.i.d.\ with a finite second moment, we prove that the left-most point of the range is located near $-u_* n^{1/3}$, where $u_* \in [0,c_h]$ is a constant that only depends on the disorder. This contrast with the homogeneous model (i.e. when $β=0$), where the left-most point has a random location between $-c_h n^{1/3}$ and $0$. With an additional moment assumption, we are able to show that the left-most point of the range is at distance $\mathcal U n^{2/9}$ from $-u_* n^{1/3}$ and the right-most point at distance $\mathcal V n^{2/9}$ from $(c_h-u_*) n^{1/3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$ are constants that depend only on $ω$. |
| title | Scaling limit of a one-dimensional polymer in a repulsive i.i.d. environment |
| topic | Probability 82B44, 60G50, 60G51 |
| url | https://arxiv.org/abs/2305.07727 |