In this work we are considering the behaviour of the limit shape of Young diagrams associated to random permutations on the set (1,...,n) under a particular class of multiplicative measures with polynomial growing cycle weights. Our method is based on generating functions and
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In this work we are considering the behaviour of the limit shape of Young diagrams associated to random permutations on the set (1,...,n) under a particular class of multiplicative measures with polynomial growing cycle weights. Our method is based on generating functions and complex analysis (saddle point method). We show that uctuations near a point behave like a normal random variable and that the joint uctuations 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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