Multi-matrix models and models with an external field

The research direction is to extend the Riemann–Hilbert/steepest-descent method to $n\times n$ Riemann–Hilbert problems, with the aim of treating generic situations in multi-matrix models and models with an external field. The intended implications include asymptotic analysis of correlation functions and connections with Padé–Hermite approximation, irrationality questions, and multiple orthogonal polynomials. This is an informal research direction, not a single quantified theorem: the class of models meant by “generic situation” and the precise asymptotic regimes are not specified. The results below describe established special cases without replacing that direction by a narrower question. Random matrices with external source and multiple orthogonal polynomials Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis

Background

Multiple orthogonality gives a link between external-source ensembles and higher-dimensional Riemann–Hilbert problems. For a Hermitian ensemble with density proportional to $\exp[-\operatorname{Tr}(V(M)-AM)]$, the average characteristic polynomial is characterized by multiple orthogonality conditions associated with the distinct eigenvalues of $A$. When $A$ has $r$ distinct eigenvalues, the corresponding Riemann–Hilbert problem has size $(r+1)\times(r+1)$. This connects matrix-model correlation kernels to multiple orthogonal polynomials, also known as Hermite–Padé polynomials. Random matrices with external source and multiple orthogonal polynomials Large n limit of Gaussian random matrices with external source, part II

The Gaussian external-source model with two source eigenvalues provided a tractable setting for extending steepest descent beyond the classical $2\times2$ problems. In the equal-multiplicity case, the associated problem is $3\times3$; the analysis uses a three-sheeted Riemann surface, lens opening, and local parametrices to control the large-$n$ limit. The two-matrix model provides a distinct, more complicated setting: its biorthogonal-polynomial formulation leads to a $4\times4$ problem in the quartic case studied below. Large n limit of Gaussian random matrices with external source Large n limit of Gaussian random matrices with external source, part II Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis

Known results

For the Gaussian external-source ensemble with $A=\operatorname{diag}(a,\ldots,a,-a,\ldots,-a)$, with each eigenvalue of multiplicity $n/2$ and $n$ even, the steepest-descent analysis of the $3\times3$ problem establishes universal local eigenvalue correlations for $a>1$: the sine kernel in the bulk and the Airy kernel at regular spectral edges. The companion analysis establishes the same bulk and edge limits for $0<a<1$, where the limiting spectrum has one interval rather than two. These are results for the specified Gaussian model and parameter ranges, not a general theorem for arbitrary external sources or arbitrary matrix sizes. Large n limit of Gaussian random matrices with external source Large n limit of Gaussian random matrices with external source, part II

A later advance beyond the $3\times3$ external-source examples is a complete steepest-descent analysis of a $4\times4$ Riemann–Hilbert problem for the two-matrix model with $W(y)=y^4/4$, $V$ an even polynomial, coupling $\tau>0$, and a one-cut regular equilibrium measure. In this setting the authors obtain global and local limiting behavior for the correlation kernel of the eigenvalues of $M_1$ after averaging over $M_2$. In particular, for $n\equiv0\pmod 3$, the regular bulk limit is the sine kernel and the regular soft-edge limit is the Airy kernel. This demonstrates that the method can handle a higher-dimensional problem in a special model, but does not provide the proposed general extension to arbitrary $n\times n$ problems. Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis