2019/07/01 by Vladislav Kargin, Kargin, Vladislav
Computer Science · Mathematics · #60C05 #60F05 #Bayesian Methods and Mixture Models #Central limit theorem #Combinatorics #Combinatorics (math.CO) #Distribution (mathematics) #Exponential distribution #FOS: Mathematics #Gaussian #Geometric distribution #Level crossing #Limit (mathematics) #Mathematical analysis #Mathematics #Partition (number theory) #Physics #Point processes and geometric inequalities #Poisson distribution #Probability (math.PR) #Probability distribution #Quantum mechanics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #math.CO #math.PR #msc:60C05 #msc:60F05
paper · pdf · doi:10.48550/arxiv.1907.00632
published in arXiv (Cornell University) (Cornell University) · 19 pages, 4 figures
arxiv created 2019/07/01 · openalex publication_date 2019/07/01 · arxiv updated 2019/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the distribution of several statistics of large non-crossing partitions. First, we prove the Gaussian limit theorem for the number of blocks of a given fixed size. In contrast to the properties of usual set partitions, we show that the number of blocks of different sizes are negatively correlated, even for large partitions. In addition, we show that the sizes of blocks in a given large non-crossing partition are distributed according to a geometric distribution and not Poisson, as in the case of usual set partitions. Next, we show that the size of the largest block concentrates at log2 n, and that after an appropriate rescaling, it can be described by the double exponential distribution. Finally, we show that the width of a large non-crossing partition converges to the Theta-distribution which arises in the theory of Brownian excursions.