This talk is based on recent joint work with Cesar Cuenca and Alex Moll. We discuss discrete analogues of a one-dimensional log-gas system with N particles in a potential V at an inverse temperature ß > 0. We establish universal formulas describing the global asymptotics of two different models of discrete ß-ensembles in high, low, and fixed-temperature regimes. These formulas exhibit surprising positivity properties and are expressed in terms of weighted lattice paths, such as Motzkin paths, Dyck paths, and, more generally, Lukasiewicz paths. Finally, we discuss the limit shape in the high/low-temperature regimes and show that, contrary to the continuous case of ß-ensembles, there is a phase transition phenomenon when transitioning from the fixed-temperature regime to the high/low-temperature regime.
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