We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy F with respect to the inverse temperature β, we show that the Hamiltonian concentrates around the energy level F’(β) under the free energy and Gibbs average.
Bibliographical noteFunding Information:
The research of the first author was partly supported by NSF Grant CAREER DMS-1653552 and NSF Grant DMS-1517894. The research of the second author was partly supported by NSF Grant DMS-1642207 and Hong Kong Research Grants Council GRF-14302515. Both authors thank the anonymous referees for the careful reading and many valuable suggestions regarding the presentation of the paper. The second author thanks the institute of mathematics at Academia Sinica and the department of applied mathematics at National Sun Yat-Sen University for their hospitality during his visit in June 2017, where part of the writing of the present paper was completed.