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## Scaling limits for the random walk penalized by its range in dimension one

Nicolas Bouchot
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#### Abstract

In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on $\mathbb{Z}$ penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by a weight $exp(−h_n |R_n |)$, with $|R_n|$ the number of visited sites and $h_n$ a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at $h_n \approx n^{1/4}$ , inherent to the underlying lattice structure.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-03579212 , version 1 (17-02-2022)
hal-03579212 , version 2 (23-02-2022)
hal-03579212 , version 3 (19-07-2022)

### Identifiers

• HAL Id : hal-03579212 , version 3
• ARXIV :

### Cite

Nicolas Bouchot. Scaling limits for the random walk penalized by its range in dimension one. 2022. ⟨hal-03579212v3⟩

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