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数学进展 - 2016, Vol. 45(6): 861-898
“纪念《数学进展》创刊六十周年”特约综述文章
随机整数划分的概率分析
Probabilistic Analysis for Random Integer Partitions

苏中根
SU Zhonggen

浙江大学数学科学学院, 杭州, 浙江, 310027
School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, 310027, P. R. China

收稿日期: 2015-01-01
出版日期: 2016-11-10
2016, Vol. 45(6): 861-898
DOI: 10.11845/sxjz.2015004a


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摘要 令$n\ge1$是自然数, 它的一个整数划分是一列自然数$\lambda_1, \lambda_2, \cdots, \lambda_l\ (l\geq 1)$使得$\lambda_1+\lambda_2+ \cdots+ \lambda_l=n$.令${\cal P}_n$表示$n$的所有整数划分的集合, $\cal P$表示所有整数划分的集合,通过赋予每个划分$\lambda$一个概率测度可以得到概率空间.特别地, 有趣的概率测度包括均匀测度和Plancherel测度以及乘性测度.这些整数随机划分的概率极限理论内容非常丰富.本文综述一些研究成果和进展, 包括极限形以及波动. 为叙述简单起见,主要集中在均匀划分和Plancherel划分,并且重点介绍两个模型中所用到条件概率的论证思想和方法,大多结果不加以详细证明.
Abstract:Let $n\ge 1$ be a natural number. An integer partition of $n$ is by definition a sequence of natural numbers $\lambda_1,\lambda_2, \cdots, \lambda_l$ ($l\ge 1$) such that $\lambda_1+\lambda_2+\cdots+\lambda_l=n$. Let ${\mathcal P}_n$ be the set of all integer partitions of $n$, ${\mathcal P} $ the set of all integer partitions. One can produce a probability space by assigning a probability to each partition $\lambda$. There are a few of interesting probability measures worthy of consideration, among which are uniform probability measure, Plancherel measure and multiplicative measure. It turns out that it is filled with a lot of nice probability limit theorems in the study of random integer partitions. In this survey we briefly review some remarkable results including limit shapes and fluctuations of a typical random partition. For sake of clarity and conciseness, we mainly concentrate on uniform integer partitions and Plancherel partitions, and the focus is upon the use of conditioning argument in both models. We omit the proofs of most results, which can be found in original articles.
PACS:  O211  
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