Master loop equations, free energy and correlations for the chain of matrices

Abstract

The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N^2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of geometric properties of the underlying algebraic curve. In particular we compute the free energy, the resolvents, the 2-loop functions and some mixed one loop functions. We also initiate the calculation of the 1/N^2 expansion.

arXiv commentlatex, 1 figure .eps. Misprints corrected, references added

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• Rational Theories of 2D Gravity from the Two-Matrix Model
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• Combinatorial Solution of the Two-Matrix Model
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Cited by in corpus (37)

• Free energy topological expansion for the 2-matrix model
• Edge Universality of Beta Ensembles
• Generation of Matrix Models by W-operators
• BGWM as Second Constituent of Complex Matrix Model
• Universality of General $\beta$-Ensembles
• Algebraic methods in random matrices and enumerative geometry
• The Cauchy two-matrix model
• Bulk Universality of General $\beta$-Ensembles with Non-convex Potential
• Large deviations and stochastic calculus for large random matrices
• Combinatorial aspects of matrix models
• Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Invariants of algebraic curves and topological expansion
• Exact 2-point function in Hermitian matrix model
• Mixed correlation functions in the 2-matrix model, and the Bethe ansatz
• Combinatorics of loop equations for branched covers of sphere
• Abstract loop equations, topological recursion, and applications
• Minimal Open Strings
• 2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals
• Loop equations for the semiclassical 2-matrix model with hard edges
• A Matrix model for plane partitions
• The complex Liouville string: the matrix integral
• Second and Third Order Observables of the Two-Matrix Model
• Topological expansion of the chain of matrices
• Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model
• A matrix model for the topological string II: The spectral curve and mirror geometry
• Topological expansion and boundary conditions
• Mirror of the refined topological vertex from a matrix model
• Multi-matrix loop equations: algebraic & differential structures and an approximation based on deformation quantization
• Non-anomalous `Ward' identities to supplement large-N multi-matrix loop equations for correlations
• Formal matrix integrals and combinatorics of maps
• Free diffusions and Matrix models with strictly convex interaction
• Schwinger-Dyson operator of Yang-Mills matrix models with ghosts and derivations of the graded shuffle algebra
• Another algebraic variational principle for the spectral curve of matrix models
• Hopf Algebraic Structure for Tagged Graphs and Topological Recursion
• Non-homogenous disks in the chain of matrices
• Solutions of loop equations are random matrices