A common way to show that anyons exhibit fractional statistics in 2D is by arguing that the paths of two anyons winding round each other cannot be continuously deformed to zero. This seems to assume that the particles cannot pass through each other. Why is this assumption valid?
We do not need to make the assumption that "the paths of two anyons winding round each other cannot be continuously deformed to zero".
To define fractional statistics, we only require that the phase of exchanging two particles do not depend on the smooth deformation of the exchange path, as long as two particles are always well separated during the exchange.
Two anyons can coincide with a finite energy cost.