Continuous Kasteleyn theory and the bead process - Samuel G. G. Johnston

We develop a continuous version of Kasteleyn theory to study the bead model on the torus. These are the point processes on the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ with the property that between every two consecutive points on same string, there lies a point on the neighbouring strings. We obtain an explicit formula for the volume of bead configurations with $n$ strings and $k$ beads per string. The asymptotics of this formula verify a recent conjecture of Shlyakhtenko and Tao (2020) in the setting of free probability. We study the correlation functions of the process, expanding on the work of Boutillier (2009). Various limits of our process recover certain Markov chains, and we show that TASEP on the ring may be constructed as an exponential change of measure of the noncolliding walkers of Gordenko (2020).