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Probabilities of rare events in product kernel aggregation: An exact formula and phase diagram

R. Goutham1,2,*, R. Rajesh1,2,†, V. Subashri3,‡, and Oleg Zaboronski4,§

  • *Contact author: gouthamr@imsc.res.in
  • †Contact author: rrajesh@imsc.res.in
  • ‡Contact author: subashriv@tifrh.res.in
  • §Contact author: olegz@maths.warwick.ac.uk
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Phys. Rev. E 114, 014152 – Published 30 July, 2026

DOI: https://doi.org/10.1103/ckzt-51sw

Abstract

We present an exact method for calculating the large deviation function describing rare fluctuations in the number of particles for product-kernel aggregation. Starting from the master equation, we derive an exact integral representation for the probability P(M,N,t) of observing N particles at time t starting from M monomers for any finite M,N,t. From this, we obtain an exact expression for the exponential moment 〈pN〉 for integer p. Employing a replica conjecture—numerically validated by finite-M scaling—we extend this result to real p≥0. The convex envelope of the large deviation function, obtained via a Legendre-Fenchel transform of the exponential moment, shows singular behavior. The singular structure allows us to construct the full phase diagram of product-kernel aggregation, which contains a tricritical point, separating continuous and discontinuous transitions. We also compute the asymptotic form of the large-deviation function for small N/M.

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References (50)

  1. G. T. Guria, M. A. Herrero, and K. E. Zlobina, A mathematical model of blood coagulation induced by activation sources, Discrete Contin. Dyn. Syst. 25, 175 (2009).
  2. G. Falkovich, A. Fouxon, and M. Stepanov, Acceleration of rain initiation by cloud turbulence, Nature (London) 419, 151 (2002).
  3. H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation: Reprinted 1980 (Springer Science & Business Media, Cham, 2012).
  4. M. Williams, A unified theory of aerosol coagulation, J. Phys. D: Appl. Phys. 21, 875 (1988).
  5. N. Brilliantov, P. Krapivsky, A. Bodrova, F. Spahn, H. Hayakawa, V. Stadnichuk, and J. Schmidt, Size distribution of particles in Saturn's rings from aggregation and fragmentation, Proc. Natl. Acad. Sci. USA 112, 9536 (2015).
  6. C. Connaughton, A. Dutta, R. Rajesh, N. Siddharth, and O. Zaboronski, Stationary mass distribution and nonlocality in models of coalescence and shattering, Phys. Rev. E 97, 022137 (2018).
  7. A. B. Burd and G. A. Jackson, Particle aggregation, Annu. Rev. Mar. Sci. 1, 65 (2009).
  8. S. Benjwal, S. Verma, K.-H. Röhm, and O. Gursky, Monitoring protein aggregation during thermal unfolding in circular dichroism experiments, Protein Sci. 15, 635 (2006).
  9. B. Wang, C. Guo, Z. Lou, and B. Xu, Following the aggregation of human prion protein on Au (111) surface in real-time, Chem. Commun. 51, 2088 (2015).
  10. A. M. Tom, R. Rajesh, and S. Vemparala, Aggregation dynamics of rigid polyelectrolytes, J. Chem. Phys. 144, 034904 (2016).
  11. A. M. Tom, R. Rajesh, and S. Vemparala, Aggregation of flexible polyelectrolytes: Phase diagram and dynamics, J. Chem. Phys. 147, 144903 (2017).
  12. A. Pineau, A. A. Benzerga, and T. Pardoen, Failure of metals I: Brittle and ductile fracture, Acta Mater. 107, 424 (2016).
  13. M. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Z. Phys. Chem. 92U, 129 (1918).
  14. D. G. Tarboton, R. L. Bras, and I. Rodriguez-Iturbe, The fractal nature of river networks, Water Resour. Res. 24, 1317 (1988).
  15. N. Berestycki, Recent progress in coalescent theory, Ensaios Matemáticos 16, 1 (2009).
  16. D. Achlioptas, R. M. D'souza, and J. Spencer, Explosive percolation in random networks, Science 323, 1453 (2009).
  17. R. M. D'Souza, J. Gómez-Gardenes, J. Nagler, and A. Arenas, Explosive phenomena in complex networks, Adv. Phys. 68, 123 (2019).
  18. F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep. 383, 95 (2003).
  19. D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists, Bernoulli 5, 3 (1999).
  20. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, UK, 2010).
  21. J. A. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: A discrete deterministic mean-field approach, Physica D 222, 1 (2006).
  22. C. Connaughton, R. Rajesh, and O. Zaboronski, Kinetics of cluster-cluster aggregation, in Handbook of Nanophysics: Clusters and Fullerenes, edited by K. D. Sattler (CRC Press, Boca Raton, FL, 2010).
  23. F. Puthalath, A. Biswas, V. V. Prasad, and R. Rajesh, Lattice models for ballistic aggregation: Cluster-shape-dependent exponents, Phys. Rev. E 108, 044127 (2023).
  24. R. Rajesh, V. Subashri, and O. Zaboronski, Exact calculation of the probabilities of rare events in cluster-cluster aggregation, Phys. Rev. Lett. 133, 097101 (2024).
  25. R. Dandekar, R. Rajesh, V. Subashri, and O. Zaboronski, A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation, Comput. Phys. Commun. 288, 108727 (2023).
  26. R. Rajesh, V. Subashri, and O. Zaboronski, Exact calculation of the large deviation function for k-nary coalescence, J. Stat. Phys. 192, 72 (2025).
  27. L. Andreis, W. König, and R. I. Patterson, A large-deviations principle for all the cluster sizes of a sparse Erdős-Rényi graph, Random Struct. Alg. 59, 522 (2021).
  28. L. Andreis, W. König, H. Langhammer, and R. I. Patterson, A large-deviations principle for all the components in a sparse inhomogeneous random graph, Probab. Theory Relat. Fields 186, 521 (2023).
  29. A. Engel, R. Monasson, and A. K. Hartmann, On large deviation properties of Erdös-Rényi random graphs, J. Stat. Phys. 117, 387 (2004).
  30. A. H. Marcus, Stochastic coalescence, Technometrics 10, 133 (1968).
  31. A. A. Lushnikov, Gelation in coagulating systems, Physica D 222, 37 (2006).
  32. A. A. Lushnikov, Evolution of coagulating systems, J. Colloid Interface Sci. 45, 549 (1973).
  33. A. A. Lushnikov, Coagulation in finite systems, J. Colloid Interface Sci. 65, 276 (1978).
  34. P. G. J. van Dongen and M. H. Ernst, Dynamic scaling in the kinetics of clustering, Phys. Rev. Lett. 54, 1396 (1985).
  35. A. A. Lushnikov, Exact kinetics of the sol-gel transition, Phys. Rev. E 71, 046129 (2005).
  36. R. C. Ball, C. Connaughton, P. P. Jones, R. Rajesh, and O. Zaboronski, Collective oscillations in irreversible coagulation driven by monomer inputs and large-cluster outputs, Phys. Rev. Lett. 109, 168304 (2012).
  37. M. Dedola and L. Cademartiri, Gelation as a dynamical instability of the smoluchowski flow, J. Phys. Chem. B 130, 4467 (2026).
  38. G. Alsmeyer, A. Bostan, K. Raschel, and T. Simon, Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials, Adv. Appl. Math. 150, 102555 (2023).
  39. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009).
  40. R. B. Griffiths and P. Gujrati, Convexity violations for noninteger parameters in certain lattice models, J. Stat. Phys. 30, 563 (1983).
  41. J. Gärtner, On large deviations from the invariant measure, Theory Probab. Appl. 22, 24 (1977).
  42. R. S. Ellis, Large deviations for a general class of random vectors, Ann. Probab. 12, 1 (1984).
  43. R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 2015).
  44. P. L. Krapivsky and S. A. Matveev, Gelation in input-driven aggregation, Phys. Rev. E 110, 034128 (2024).
  45. R. R. Dyachenko, S. A. Matveev, and P. L. Krapivsky, Finite-size effects in addition and chipping processes, Phys. Rev. E 108, 044119 (2023).
  46. J. L. Spouge, Exact solutions for a diffusion-reaction process in one dimension, Phys. Rev. Lett. 60, 871 (1988).
  47. K. Kang and S. Redner, Fluctuation effects in Smoluchowski reaction kinetics, Phys. Rev. A 30, 2833 (1984).
  48. S. Krishnamurthy, R. Rajesh, and O. Zaboronski, Kang-Redner small-mass anomaly in cluster-cluster aggregation, Phys. Rev. E 66, 066118 (2002).
  49. S. Krishnamurthy, R. Rajesh, and O. Zaboronski, Persistence properties of a system of coagulating and annihilating random walkers, Phys. Rev. E 68, 046103 (2003).
  50. R. B. Ravindran and R. Rajesh, Finite-size scaling and edge effects in the Takayasu model of aggregation-diffusion with input, Phys. Rev. E 113, 054135 (2026).

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Phys. Rev. E 114, 014152– Published 30 July, 2026
  • Received 4 February 2026
  • Revised 19 May 2026
  • Accepted 10 July 2026
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