The limit shape of random permutations with polynomially growing cycle weights

Abstract

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots ,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.

36 pages, 3 figures. The paper was subject to a major revision (compared to v1): 1) we considered more general weights, i. e. ${\theta }_m=(\log m{)}^j{m}^{\alpha }$, 2) title replaced, 3) improvements of the presentation, 4) correction of typos and minor mathematical errors

Cited by in corpus (3)

• Random partitions in statistical mechanics
• Limit shapes for Gibbs ensembles of partitions
• Limit shapes for Gibbs partitions of sets