The bead process on the torus
Consider the semi-discrete torus T_n := [0, 1) × {0, 1, . . . , n − 1} representing n unit length strings running in parallel. A bead configuration is a point process on T_n with the property that between every two consecutive points on same string, there lies a point on each of the neighbouring strings.
We develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on T_n may be expressed in terms of Fredholm determinants of certain operators on T_n, and thereby obtain an explicit formula for the volume of bead configurations. The asymptotics of this volume formula confirm a recent prediction of Shlyakhtenko and Tao in the free probability literature.
Thereafter we study random bead configurations on T_n, making connections with exclusion processes and providing a new probabilistic representation of TASEP on the ring.