The limit shape of random permutations with polynomially growing cycle weights

2013/12/12 by Alessandra Cipriani, Cipriani, Alessandra, Dirk Zeindler +1
Mathematics · #60F05 #60F10 #60F17 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F05 #msc:60F10 #msc:60F17

paper · pdf · doi:10.48550/arxiv.1312.3517

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

arxiv created 2014/07/09 · arxiv updated 2014/07/10

Abstract

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set \1,…,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.

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