The spectrum of local random Hamiltonians

The spectrum of a local random Hamiltonian can be represented generically by the so-called ϵ -free convolution of the probability distributions of its local terms. We establish an isomorphism between the set of ϵ -noncrossing partitions and permutations to study its spectrum. Moreover, we derive some lower and upper bounds for the maximal eigenvalue of the Hamiltonian.