It is often argued that thermodynamic ensembles are equivalent in the sense that no matter what ensemble one uses for the calculations, one should end up in the same macroscopic equations of state. This is due to the fact that distributions are sharply peaked around average values and are essentially Dirac delta functions around averages.
Some quantities, however, are related to fluctuations. For example, the specific heat is related to energy fluctuations via the fluctuation-dissipation theorem. I would expect that, in this case, the actual shape of the peaked distribution function matters because that's what describes the fluctuations. How is it then possible that calculating fluctuation related quantities, such as specific heats or magnetic susceptibility, is still ensemble-independent?