Dynamical Loop Equation

We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two–dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, $\beta$–corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and $(q,\kappa )$-distributions on lozenge tilings. Under technical assumptions, we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an application, we compute the limit shape for $(q,\kappa )$–distributions on lozenge tilings and prove that their height fluctuations converge to the Gaussian Free Field in an appropriate complex structure.