LAUSR

Building on our earlier work \unscite{Sa-Za}, we introduce and study generalized XX0 models. We explicitly construct a long-range interacting spin chain, referred to as the Selberg model, and study the correlation functions of the Selberg and XX0 models. Using a matrix integral representation of the generalized XX0 model and applying asymptotic analysis in non-intersecting Brownian motion, the phase structure of the Selberg model is determined. We find that tails of the Tracy-Widom distribution, of Gaussian unitary ensemble, govern a discrete-to-continuous third-order phase transition in Selberg model. The same method also reproduces the Gross-Witten phase transition of the original XX0 model. Finally, we conjecture universal features for the phase structure of the generalized XX0 model.