  We consider random permutations on $\Sn$ with logarithmic growing cycles
weights and study asymptotic behavior as the length $n$ tends to infinity. We
show that the cycle count process converges to a vector of independent Poisson
variables and also compute the total variation distance between both processes.
Next, we prove a central limit theorem for the total number of cycles.
Furthermore we establish a shape theorem and a functional central limit theorem
for the Young diagrams associated to random permutations under this measure.
  We prove these results using tools from complex analysis and combinatorics.
In particular we have to apply the method of singularity analysis to generating
functions of the form $\exp\left( (-\log(1-z))^{k+1} \right)$ with $k\geq 1$,
which have not yet been studied in the literature.
