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On the random Young diagrams and their cores. (English) Zbl 0929.05088

The authors consider Young diagrams or Ferrers graphs that are chosen uniformly at random from among all partitions of \(n\). They determine the expected shape and size of the \(r\)-core. In particular, when scaled by \(n^{1/2}\), the core size is asymptotically gamma-distributed with parameter \((r-1)/2\). For \(n\) chosen uniformly at random from 1 to \(N\) and the core diagram scaled both horizontally and vertically by \(N^{1/4}\), the boundary converges to a random convex polygonal curve with \(r-1\) straight line segments.

MSC:

05E10 Combinatorial aspects of representation theory
05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
60C05 Combinatorial probability
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