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

2022/02/24 by Nicolas Bouchot, Bouchot, Nicolas · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2202.11953

openalex publication_date 2022/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on 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 ≈ n^1/4 , inherent to the underlying lattice structure.

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• Non-directed polymers in random environments with range penalties: the high dimensional case

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