Dynamical loop equation (Q6618732)

The authors introduce general loop equations for two-dimensional interacting particle systems where the transition probabilities of the one-dimensional systems with \(n\) and \(n+1\) particles are related by corresponding interlacing properties (called descending transitions in the paper). It is shown that the loop equations of the authors fit to many known particle models like Dyson Brownian motion, nonintersecting Bernoulli and Poisson random walks, \(\beta\)-corner processes, uniform and Jack-type Gelfand-Tsetlin patterns, models associated with Macdonald and Koornwider polynomials, as well as distributions of lozenge tilings. The authors present estimates for means and covariances and show that their loop equations lead to Gaussian field type fluctuations under some technical conditions. In particular, in the tiling cases, the limit shape and a law of large numbers are derived where the fluctuations of the heights converge to the Gaussian free field.