Emergence of generic first-passage time distributions for large Markovian networks
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912979865305088 |
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| author | Voits, Julian B. Schwarz, Ulrich S. |
| author_facet | Voits, Julian B. Schwarz, Ulrich S. |
| contents | First-passage times are often the most relevant aspect of a complex Markovian network, because they signify when information processing has resulted in a definite decision. Previous studies have shown that for kinetic proofreading networks in the limit of large network size the first-passage time distribution converges either to a delta or to an exponential distribution. Remarkably, these two forms correspond to the two extreme distributions of minimal and maximal entropy for a fixed mean, respectively. Here we build on the connection between first-passage times and graph theory to show that these two limits are not model-specific, but arise generically in Markovian networks from the distribution of the eigenvalues of the generator matrix. A deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue. We also show that the exponential limit emerges robustly for reversible networks when a backward bias exists. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but under structurally tighter conditions, revealing a fundamental asymmetry between both regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18265 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Emergence of generic first-passage time distributions for large Markovian networks Voits, Julian B. Schwarz, Ulrich S. Statistical Mechanics Biological Physics Molecular Networks First-passage times are often the most relevant aspect of a complex Markovian network, because they signify when information processing has resulted in a definite decision. Previous studies have shown that for kinetic proofreading networks in the limit of large network size the first-passage time distribution converges either to a delta or to an exponential distribution. Remarkably, these two forms correspond to the two extreme distributions of minimal and maximal entropy for a fixed mean, respectively. Here we build on the connection between first-passage times and graph theory to show that these two limits are not model-specific, but arise generically in Markovian networks from the distribution of the eigenvalues of the generator matrix. A deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue. We also show that the exponential limit emerges robustly for reversible networks when a backward bias exists. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but under structurally tighter conditions, revealing a fundamental asymmetry between both regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks. |
| title | Emergence of generic first-passage time distributions for large Markovian networks |
| topic | Statistical Mechanics Biological Physics Molecular Networks |
| url | https://arxiv.org/abs/2602.18265 |