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Exact asymptotic equivalence between precisions of record-times and integrated currents

  • Alberto Garilli
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Abstract

We establish an exact equivalence between the fluctuations of integrated currents and the temporal statistics of current record events in continuous-time Markov jump processes. Specifically, we show that, under minimal assumptions, the Fano factor of an integrated current coincides exactly with the squared coefficient of variation of the inter-record waiting times. This result is obtained by showing that inter-record times induce a renewal structure, since each record event resets the system to a fixed state, and that their generating function can be expressed exactly in terms of a combinatorial organization governed by Narayana numbers. The equivalence extends previous results obtained for unicyclic networks to arbitrary finite-state graphs and provides a unified representation of current fluctuations in terms of record-time observables. The results are derived within a purely probabilistic framework and clarify the structural origin of current precision in terms of first-passage-like event statistics.

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Cite

BibTeXRIS
BibTeX
@misc{garilli2026exact,
  title = {{Exact asymptotic equivalence between precisions of record-times and integrated currents}},
  author = {Alberto Garilli},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09479},
  eprint = {2609.09479},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09479},
}
RIS
TY  - UNPB
TI  - Exact asymptotic equivalence between precisions of record-times and integrated currents
AU  - Alberto Garilli
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09479
UR  - https://arxiv.org/abs/2609.09479
AB  - We establish an exact equivalence between the fluctuations of integrated currents and the temporal statistics of current record events in continuous-time Markov jump processes. Specifically, we show that, under minimal assumptions, the Fano factor of an integrated current coincides exactly with the squared coefficient of variation of the inter-record waiting times. This result is obtained by showing that inter-record times induce a renewal structure, since each record event resets the system to a fixed state, and that their generating function can be expressed exactly in terms of a combinatorial organization governed by Narayana numbers. The equivalence extends previous results obtained for unicyclic networks to arbitrary finite-state graphs and provides a unified representation of current fluctuations in terms of record-time observables. The results are derived within a purely probabilistic framework and clarify the structural origin of current precision in terms of first-passage-like event statistics.
ER  -