Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912351039520768 |
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| author | Blancas, Airam Fittipaldi, Maria Clara Hernandez-Torres, Sarai |
| author_facet | Blancas, Airam Fittipaldi, Maria Clara Hernandez-Torres, Sarai |
| contents | The mother-dependent neutral mutations model describes the evolution of a population across discrete generations, where neutral mutations occur among a finite set of possible alleles. In this model, each mutant child acquires a type different from that of its mother, chosen uniformly at random. In this work, we define a multitype allele tree associated with this model and analyze its scaling limit through a Markov chain that tracks the sizes of allelic subfamilies and their mutant descendants. We show that this Markov chain converges to a continuous-state Markov process, whose transition probabilities depend on the sizes of the initial allelic populations and those of their mutant offspring in the first allelic generation. As a result, the allele tree converges to a multidimensional limiting object, which can be described in terms of the universal allele tree introduced by Bertoin (2010). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime Blancas, Airam Fittipaldi, Maria Clara Hernandez-Torres, Sarai Probability 60J80, 92D25 The mother-dependent neutral mutations model describes the evolution of a population across discrete generations, where neutral mutations occur among a finite set of possible alleles. In this model, each mutant child acquires a type different from that of its mother, chosen uniformly at random. In this work, we define a multitype allele tree associated with this model and analyze its scaling limit through a Markov chain that tracks the sizes of allelic subfamilies and their mutant descendants. We show that this Markov chain converges to a continuous-state Markov process, whose transition probabilities depend on the sizes of the initial allelic populations and those of their mutant offspring in the first allelic generation. As a result, the allele tree converges to a multidimensional limiting object, which can be described in terms of the universal allele tree introduced by Bertoin (2010). |
| title | Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime |
| topic | Probability 60J80, 92D25 |
| url | https://arxiv.org/abs/2504.20030 |