Abstract
This is the second paper in a series studying the global asymptotics of discrete $N$-particle systems with inverse temperature parameter $\theta$ in the high temperature regime. In the first paper, we established necessary and sufficient conditions for the Law of Large Numbers at high temperature in terms of Jack generating functions. In this paper, we derive a functional equation for the moment generating function of the limiting measure, which enables its analysis using analytic tools. We apply this functional equation to compute the densities of the high temperature limits of the pure Jack measures. As a special case, we obtain the high temperature limit of the large fixed-time distribution of the discrete-space $\beta$-Dyson Brownian motion of Gorin–Shkolnikov. Two special cases of our densities are the high temperature limits of discrete versions of the G$\beta$E, computed by Allez-Bouchaud-Guionnet in [Phys. Rev. Lett. 109 (2012), 094102; arXiv:1205.3598], and LβE, computed by Allez-Bouchaud-Majumdar-Vivo in [J. Phys. A, vol. 46, no. 1 (2013), 015001; arXiv:1209.6171]. Moreover, we prove the following crystallization phenomenon of the particles in the high temperature limit: the limiting measures are uniformly supported on disjoint intervals with unit gaps and their locations correspond to the zeros of explicit special functions with all roots located in the real line. We also show that these zeros correspond to the spectra of certain unbounded Jacobi operators.