Anyonic Partition Functions and Windings of Planar Brownian Motion

The computation of the $N$-cycle brownian paths contribution $F_N(\alpha)$ to the $N$-anyon partition function is adressed. A detailed numerical analysis based on random walk on a lattice indicates that $F_N^{(0)}(\alpha)= \prod_{k=1}^{N-1}(1-{N\over k}\alpha)$. In the paramount $3$-anyon case, one can show that $F_3(\alpha)$ is built by linear states belonging to the bosonic, fermionic, and mixed representations of $S_3$.