Ben Arous, GerardGayrard, VeroniqueKuptsov, Alexey2010-11-302010-11-302010-11-30200810.1007/978-3-7643-8786-0_4https://infoscience.epfl.ch/handle/20.500.14299/61044WOS:000259173100004We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit, as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the P-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down.statistical mechanicsdisordered mediaspin-glassesEnergytext::conference output::conference proceedings::conference paper