Total variation distance and the Erd\H{o}s-Tur\'an law for random permutations with polynomially growing cycle weights

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 $1,2,\mathrm{.}\mathrm{.}\mathrm{.},b$ and a process $(Z_1,Z_2,\mathrm{.}\mathrm{.}\mathrm{.},Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\mathrm{\ell })$ where $\mathrm{\ell }$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erd\H{o}s-Tur'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.