The Fluctuations Of Random Partitions Of Naturals With Fixed Number Of Summands

An integer partition means dividing a positive integer into a sum of positive integers and we take no account of the order of these integers.The number of different partition of the positive integer n is called the partition number of the n.If we assign a probability measure to each partition of an integer,this naturally creates a probability space.We can equip a variety of measures.Depending on different measure,some results will be different.When equipped with uniform measures,In 1952 Tempeley first discovered that there is a weak law of large numbers(limiting shapes)for integer partition.The rigorous argument was given by Vershik and Kerov in 1985.Based on the ideas of Vershik,the problem of the limiting shapes was developed in many directions.We consider the uniform distribution on the set of partitions of integer n with(?)numbers of summands,c>0 is a positive constant.On this premise,Vershik and Yakubovich gave limiting shapes(a weak law of large numbers)in 2001.Based on limiting shapes and other asymptotic properties studied by Vershik and Yakubovich,we establish central limit theorems for the fluctuations of parts around the limiting shapes at the edge.Our proof consists of two parts:attack the corresponding problems with independence in a proper grand ensemble and transfer the results to a small ensemble by conditional probability.Our proof draws on the idea which is first applied in the context of random partitons by Fristedt in 1993.The idea aims to introduce a suitable measure μx,y in the grand ensemble,such that a given uniform measure μ(n,m)is just viewed as the conditional distribution of μx,y In introducing suitable measures,this paper draws on the papers of Vershik and Yakubovich,which are based on the multiplicative measure first defined by Vershik.The framework of this thesis is organized as follows:In the first part,we introduce the development history of integer partition and related research content.In the second part,we introduce the relevant knowledge covered in this paper.In the third part,we divide it into three subsections.In subsection 3.1 we give a brief introduction to the paper.In subsection 3.2 we first give the expectation and variance of some random variables on the grand ensemble.And then prove the main result holds on the grand ensemble.To transfer the results from grand ensemble to small ensemble,we give Lemmas 3.2-3.4.Finally,using the above lemmas and Theorem 3.3 in subsection 3.2,we finish the proofs of the main results of this paper.