Anyonic Partition Functions and Windings of Planar Brownian Motion

TL;DR

This paper analyzes the contribution of N-cycle Brownian paths to N-anyon partition functions, providing numerical insights and exploring the structure of the 3-anyon case through group representations.

Contribution

It introduces a detailed numerical approach to compute N-anyon partition functions and reveals the structure of the 3-anyon case using group representations.

Findings

Numerical analysis shows $F_N^{(0)}( ext{alpha})$ as a product over k

In the 3-anyon case, $F_3( ext{alpha})$ involves bosonic, fermionic, and mixed states

The study connects Brownian motion windings with anyonic statistics

Abstract

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 $S_3$.