Breakdown of universality in multi-cut matrix models

Abstract

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random N × N matrices with a disconnected eigenvalue support. We show that the difference does not stem from a Z 2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean field expressions. Our result invalidates the existence of a smooth topological large N expansion and some postulated universality properties of correlators. We derive the large N expansion of the free energy for the general 2-cut case. From it we rederive by a direct and easy meanfield-like method the 2-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.

A-polynomial, B-model, and quantization

“…, a 4 , and the corresponding two-point function has been found, using matrix model techniques, in [36]. This result can also be obtained, see [37], using properties of elliptic functions and rewriting the Bergman kernel given above, so that 20…”

Gravitational corrections in supersymmetric gauge theory and matrix models

“…For this purpose it is convenient to change variables, as in [19], (x 1 , x 2 , x 3 , x 4 ) → (∆ 21 , ∆ 43 , Q, I), where the explicit relations were given in (2.18). 4 The complete matrix-model partition function involves a sum over all possible eigenvalue distributions and does not have a topological expansion, as explained in [33]. This subtlety is, however, not relevant here as we fix N 1 and N 2 .…”

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

“…In the 1-matrix case [10,11], this was provided by means of hyperelliptic Θ-functions. The physical heuristics and the basic tools for generating such ansatz were also given in [7,14,23,18]. Much of of these heuristics can be extended to the 2-matrix case [4], and this will be the subject of a subsequent work [1].…”

“…Let L be the dual sector, i.e., the sector centered around π − α 0 (= 0) with width π − ǫ. Any of the functions Ψ and the sector R contains the sectors S (7,9,11) x . Finally, fix a contour Γ which goes off to infinity asymptotic to the two rays ℓ L , ℓ R (as in figure).…”