Persistence exponents of self-interacting random walks
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866918064938811392 |
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| author | Brémont, Julien Régnier, Léo Bénichou, Olivier Voituriez, Raphaël |
| author_facet | Brémont, Julien Régnier, Léo Bénichou, Olivier Voituriez, Raphaël |
| contents | The persistence exponent, which characterises the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. Determining this exponent for non-Markovian processes is known to be a difficult task, and exact results remain scarce despite sustained efforts. In this Letter, we consider the fundamental class of self-interacting random walks (SIRWs), which display long-range memory effects that result from the interaction of the random walker at time $t$ with the territory already visited at earlier times $t'<t$. We compute exactly the persistence exponent for all physically relevant SIRWs. As a byproduct, we also determine the splitting probability of these processes. Besides their intrinsic theoretical interest, these results provide a quantitative characterization of the exploration process of SIRWs, which are involved in fields as diverse as foraging theory, cell biology, and machine learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Persistence exponents of self-interacting random walks Brémont, Julien Régnier, Léo Bénichou, Olivier Voituriez, Raphaël Statistical Mechanics Probability The persistence exponent, which characterises the long-time decay of the survival probability of stochastic processes in the presence of an absorbing target, plays a key role in quantifying the dynamics of fluctuating systems. Determining this exponent for non-Markovian processes is known to be a difficult task, and exact results remain scarce despite sustained efforts. In this Letter, we consider the fundamental class of self-interacting random walks (SIRWs), which display long-range memory effects that result from the interaction of the random walker at time $t$ with the territory already visited at earlier times $t'<t$. We compute exactly the persistence exponent for all physically relevant SIRWs. As a byproduct, we also determine the splitting probability of these processes. Besides their intrinsic theoretical interest, these results provide a quantitative characterization of the exploration process of SIRWs, which are involved in fields as diverse as foraging theory, cell biology, and machine learning. |
| title | Persistence exponents of self-interacting random walks |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2410.18699 |