Vicious walkers and random contraction matrices

The ensemble ${\text{CUE}}^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q\mathrm{.}$ ${\text{CUE}}^{(q)}$ is an exactly solved model of random contraction matrices originally introduced in the context of scattering theory. In this article, we exhibit a connection between ${\text{CUE}}^{(q)}$ and Fisher’s random-turns vicious walker model from statistical mechanics. In particular, we show that the moment generating function of the trace of a random matrix from ${\text{CUE}}^{(q)}$ is a generating series for the partition function of Fisher’s model, when the walkers are assumed to represent mutually attracting particles.