Limit distributions for cycles of random parking functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917358352728064 |
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| author | Paguyo, J. E. Yin, Mei |
| author_facet | Paguyo, J. E. Yin, Mei |
| contents | We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the scaled number of cyclic points of a random parking function is asymptotically Rayleigh distributed; we establish the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles in a random parking function; we also compute the asymptotic mean of the length of the $r$th longest cycle in a random parking function for all valid $r$. A variety of tools from probability theory and combinatorics are used in our investigation. Corresponding results for the class of prime parking functions are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07110 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit distributions for cycles of random parking functions Paguyo, J. E. Yin, Mei Probability Combinatorics 60C05, 60F05 We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the scaled number of cyclic points of a random parking function is asymptotically Rayleigh distributed; we establish the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles in a random parking function; we also compute the asymptotic mean of the length of the $r$th longest cycle in a random parking function for all valid $r$. A variety of tools from probability theory and combinatorics are used in our investigation. Corresponding results for the class of prime parking functions are obtained. |
| title | Limit distributions for cycles of random parking functions |
| topic | Probability Combinatorics 60C05, 60F05 |
| url | https://arxiv.org/abs/2502.07110 |