Law of Large Numbers for continuous $N$-particle ensembles at fixed temperature

In this paper, we find necessary and sufficient conditions for the Law of Large Numbers of averaged empirical measures of $N$-particle ensembles, in terms of the asymptotics of their Bessel generating functions, in the fixed temperature regime. This settles an open problem posed by Benaych-Georges, Cuenca and Gorin. For one direction, we use the moment method through Dunkl operators, and for the other we employ a special case of the formula of Chapuy–Dolega for the generating function of infinite constellations. As applications, we prove that the LLN for $\theta$-sums and $\theta$-corners of random matrices are given by the free convolution and free projection, respectively, regardless of the value of inverse temperature parameter $\theta$. We also prove the LLN for a time-slice of the $\theta$-Dyson Brownian motion.