Fluctuating ensemble averages and the BTZ threshold

Recent work shows fascinating links between ensemble averaging and the Swampland program. In order to break the emerging global symmetries of the ensemble averaging as dictated by the no global symmetries conjecture, one must consider fluctuations away from the average given by deviations in the Siegel–Weil formula. In this work, we investigate the physical interpretation of these fluctuations in the bulk physics and pinpoint the states giving rise to them. For this purpose, we explore an ensemble of generalised Narain CFTs and build the AdS\(_{3}\) gravitational dual in the Chern–Simons (CS) framework. We study the associated charged BTZ black hole solution and assess its stability. Using the Swampland weak gravity conjecture, we show that the fluctuations of the ensemble average are below the BTZ threshold and correspond to a sublattice of superextremal states emitted by the black hole. We exploit the logarithmic density of states to derive bounds on the charged vectors of the abelian CS symmetry and introduce a novel formulation of the density function to ensure consistency with the sublattice WGC. We establish bounds that allows to distinguish heavy states contributing to the average from light states generating fluctuations around it.