On the statistical mechanics approach in the random matrix theory: Integrated density of states

On the Law of Addition of Random Matrices

“…[5]) in the same way as for the GUE (2.28), where V (λ) = λ 2 /4w 2 (see [18]). Condition (2.17) follows from results of [6,22]. Thus we can apply Theorem 2.1 to obtain rigorously relation (2.36) in the case when matrices H r , r = 1, 2 in (2.1) are distributed according to (2.37).…”

“…The derivation is based on the perturbation theory with respect to the non-quadratic part of V 1,2 and the R-transform is related to the sum of irreducible diagrams of the formal perturbation series. Existence of the limiting eigenvalue counting measure for the random matrix ensemble (2.37) was rigorously proved in [6] for a rather broad class of functions V (not necessary polynomials). It was also proved that the normalized counting measure (2.2) converges in probability to the limiting measure.…”

Bulk Universality and Related Properties of Hermitian Matrix Models

“…In statistical mechanics terms we have here a mean field model in which the temperature is inverse proportional to the number of particles, while in a standard statistical mechanics treatment the temperature is fixed during the "macroscopic limit" n → ∞. This will imply that the free energy of the model has to be divided by n 2 to have a well defined limit as n → ∞ and that the limit will coincide with the limit as n → ∞ of the ground state energy, also divided by n 2 (see [1,14] and formulas (2.10) -(2.11), and (2.28) below).…”

“…In Section 2 we treat the global regime and in Section 3 the local regime. In the course of our presentation we will need several technical results from [1,21]. We will give them here (often improving) to make the paper self consistent.…”

“…We obtain the sin-kernel as a unique solution of a certain nonlinear integro-differential equation, while in our previous paper [21] the kernel was obtained, roughly speaking, as a power series in its arguments. Since our proof of universality requires a number of facts on limiting Normalized Counting Measure of eigenvalues of matrix models, the paper includes an updated and simplified version of results of [1] on the existence and properties of the measure. Most of simplifications are possible because of systematic use of book [22] The paper is organized as follows.…”