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Discrete mathematical modelling seminar
Seminar information archive ～10/27｜Next seminar｜Future seminars 10/28～
| Organizer(s) | Tetsuji Tokihiro, Ralph Willox |
|---|
2026/11/26
17:00-18:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Mariusz Białecki (Polish Academy of Sciences・Institute of Geophysics)
Catalan-like sequences as limits of a stochastic aggregation-disintegration process (English)
Mariusz Białecki (Polish Academy of Sciences・Institute of Geophysics)
Catalan-like sequences as limits of a stochastic aggregation-disintegration process (English)
[ Abstract ]
We define a stochastic process using simple, parameter-dependent rules for the aggregation of entities with the possibility of disintegration of the formed clusters and derive the corresponding Smoluchowski-type equations. We show that in the limit of infinite system size, the stationary distribution of cluster sizes is given by a solvable recurrence that, after rescaling, takes the form of a Catalan-like recurrence. This recurrence, for a specific choice of parameters, reproduces various integer sequences, including the Catalan, Motzkin, and Schröder numbers. We also solve the inverse problem, i.e., we show how to determine the values of the process's probability parameters to obtain a recurrence with predetermined coefficients.
We define a stochastic process using simple, parameter-dependent rules for the aggregation of entities with the possibility of disintegration of the formed clusters and derive the corresponding Smoluchowski-type equations. We show that in the limit of infinite system size, the stationary distribution of cluster sizes is given by a solvable recurrence that, after rescaling, takes the form of a Catalan-like recurrence. This recurrence, for a specific choice of parameters, reproduces various integer sequences, including the Catalan, Motzkin, and Schröder numbers. We also solve the inverse problem, i.e., we show how to determine the values of the process's probability parameters to obtain a recurrence with predetermined coefficients.
