Breakdown of universality in multi-cut matrix models

By G. Bonnet, F. David, B. Eynard. 2000

177 citations recorded by the source corpus.

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 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 two-cut case. From it we rederive by a direct and easy mean-field-like method the two-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/02ca37348de1e3990efcedc72718eb16a8cc90d3

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Citations in the corpus

Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models (2009, background): “uch that the integral becomes convergent. Indeed, in general, the various phases of matrix models are separated by singular domains (in the space of complex potentials), where no large N limit exists [2, 3, 88]. Removal of the “divergent regions” where ReV (z) → −∞ as |z| → ∞ corresponds to holes in”
Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings (2012, background): “In the multi-cut case one has to do a slightly refined analysis, see the Appendix in the published version of [27] for an explicit derivation.”
Les Houches lectures on matrix models and topological strings (2004, background): “Usually, the extra condition which is imposed is that the different cuts are at equipotential lines (see for example [11, 4]).”
Topological Strings from Quantum Mechanics (2014, methodology): “73) The theta function gives an oscillatory, non-perturbative correction which is similar to the oscillatory corrections to the large N asymptotics of matrix models [54].”
Loop equations for the semiclassical 2-matrix model with hard edges (2005, background): “The Y (x) which is the large N limit of the resolvent is the one for which R(F0) is minimal (see [15, 4]).”
Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings (2007, background): “Some aspects of this framework were discussed in [67], albeit in a different context.”
Perturbative analysis of gauged matrix models (2002, unknown): “) This term can be exponentiated directly in the action (see also [15]) giving the effective action 2 tr log (Φ11 ⊗ 1− 1⊗ Φ22) .”
Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, background): “6.1 General picture 41 6.2 A toy model 42 6.3 Holographic description and nonperturbative effects 44”
Matrix model as a mirror of Chern-Simons theory (2002, methodology): “Many of the techniques developed here extend to more general non-compact Calabi-Yau geometries.”
Open string amplitudes and large order behavior in topological string theory (2006, unknown): “29”
Hermitian matrix model free energy: Feynman graph technique for all genera (2005, background): “These solutions, being known already for a long time [4, 5], got a new insight in [6].”
The Resurgence of Instantons in String Theory (2011, background): “This naturally led to the construction of a grand–canonical, manifestly background independent, partition function in [27] (building upon results in [28, 29]) which was further proved to be both holomorphic and modular covariant.”
A primer on resurgent transseries and their asymptotics (2018, background): “, modular properties [46, 111, 112, 179, 17, 18], transasymptotics [72, 73] and its “nonlinear” generalizations [17, 18] (all relating back to modularity [46, 111, 112]), and (generalized) partition-function phases [17, 18].”
A-polynomial, B-model, and quantization (2011, methodology): “This result can also be obtained, see [39], using properties of elliptic functions and rewriting the Bergman More generally, one can consider a generalized Bergman kernel [12], which differes from an ordinary Bergman kernel by a dependence on an additional parameter κ.”
Gravitational corrections in supersymmetric gauge theory and matrix models (2002, background): “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].”