Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts

作者：Bogachev Leonid V^*

来源：Random Structures and Algorithms, 2015, 47(2): 227-266.

DOI：10.1002/rsa.20540

• integer partitions
• Young diagrams
• limit shape
• local limit theorem
• generating functions
• cumulants

We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form F(z)=<mml:mstyle displaystyle="true"><mml:munderover>=1</mml:munderover>F0</mml:mstyle>(z) (which entails equal weighting among possible parts N). Under mild technical assumptions on the function H0(u)=ln(F0(u)), we show that the limit shape *(x) exists and is given by the equation y=-1H0(e-x), where 2=<mml:mstyle displaystyle="true">01</mml:msubsup>u-1</mml:mstyle><mml:msub>H0(u)du. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta-types of decomposable combinatorial structures assemblies, multisets, and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.

• 出版日期2015-9