The aim of the talk is to provide a quick introduction into probability methods in the asymptotic theory of partitions of natural numbers. We will briefly describe the beautiful approach proposed by Sinai and Vershik in 1990s and will show how it can be used to obtain Hardy--Ramanujan's asymptotic formula for the partition function. We will also sketch the derivation of limit shape for Young diagrams representing partitions (under the uniform distribution). Classes of partitions with various restrictions (e.g. with different summands) can also be studied by this method. An analogy with statistics of ideal quantum gas (bosons vs. fermions) will be pointed out. An intermediate "anyonic" case will also be discussed in some detail.
The talk will survey constructions and properties of ordered and unordered partitions induced by sampling, in particular various generalisations of the Ewens-Pitman two-parameter family of partitions.
In this talk I will explain some connections between partitions and compositions and their limit shapes when their weight and length grow in a certain regime. More exactly, we consider a uniform measure on partitions of weight n and length m . For n, m→∞ with m3 = o(n) it was shown in 1940s by Erdös and Lehner that there are approximately m! times more compositions than partitions. It implies that many properties of the uniform measures are asymptotically the same for compositions and partitions growing in this regime.
I will show that the last statement can be extended to the growth satisfying m2 = o(n). This will be explained via a notion of "sliced" Young diagrams.
See PDF-file.