Abstract
This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of n which maximize the number of subpartitions. The limit shape and the growth rate of the number of subpartitions are explicit. The key ideas are to use large deviations estimates for random walks, together with convex analysis and the Hardy–Ramanujan asymptotics. Our limit shape coincides with Vershik’s limit shape for uniform random partitions.
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Acknowledgements
The authors are thankful to Greg Martin and Richard Stanley who initiated a conversation on MathOverflow 2 years ago on this question, and in particular to Richard Stanley who posed this question to the first author of this work. I. Corwin was partially supported by a Packard Foundation Science and Engineering Fellowship as well as NSF Grants DMS:1811143 and DMS:1664650. S. Parekh was partially supported by the Fernholz Foundation’s “Summer Minerva Fellows” Program, as well as summer support from I. Corwin’s NSF Grant DMS:1811143.
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Communicated by Herbert Spohn.
This note is dedicated to Joel Lebowitz in appreciation for his tremendous and ongoing contributions to the world of statistical physics.
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Corwin, I., Parekh, S. Limit Shape of Subpartition-Maximizing Partitions. J Stat Phys 180, 597–611 (2020). https://doi.org/10.1007/s10955-019-02481-3
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DOI: https://doi.org/10.1007/s10955-019-02481-3