Breakdown of universality in multi-cut matrix models
Gabrielle Bonnet, Francois David, Bertrand Eynard
TL;DR
This work resolves the apparent mismatch between mean-field predictions and orthogonal-polynomials analyses in multi-cut random matrix ensembles by isolating the source of discrepancy: the integer nature of eigenvalue occupation numbers induces quasiperiodic corrections in $N$, obstructing a smooth $1/N^2$ topological expansion. By performing a detailed two-cut analysis, the authors derive the large-$N$ free-energy including the discrete-theta corrections, obtain the complete two-point correlator with both universal and non-universal terms, and provide the exact asymptotics for the orthogonal polynomials via the kernel. The non-regular, $N$-dependent terms are shown to be expressible through elliptic functions, tying the results to the modular parameter $\tau$ of the associated torus. These findings extend to any number of cuts and to complex potentials, establishing that short-range correlations remain universal while long-range correlations acquire $N$-dependent quasi-periodicity, thereby refining universality notions in multi-cut ensembles.
Abstract
We solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random NxN matrices with a disconnected eigenvalue support. We show that the difference does not stem from a Z2 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 mean-field-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.