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We study the model of random permutations of $n$ objects with polynomiallygrowing cycle weights, which was recently considered by Ercolani and Ueltschi,among others. Using saddle-point analysis, we prove that the total variationdistance between the process which counts the cycles of size $1, 2, ..., b$ anda process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variablesconverges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length ofa typical cycle in this model. By means of this result, we prove a centrallimit theorem for the order of a permutation and thus extend theErd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motionlimit theorem for the small cycles.

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