Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

@inproceedings{Storm2014TotalVD,
  title={Total variation distance and the Erdős–Tur{\'a}n law for random permutations with polynomially growing cycle weights},
  author={Jesper Storm and Dirk Zeindler},
  year={2014}
}
We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By… CONTINUE READING