Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration. (English) Zbl 1140.82307

Ercolani, N. M.; McLaughlin, K. D. T.-R.

Int. Math. Res. Not. 2003, No. 14, 755-820 (2003).

The authors establish three new theorems (Theorems 1.1, 1.3, 1.4) on asymptotics for the following family of integrals \[ Z_N(t_1,t_2,\dots, t_\nu)\overset{\text{def}} =\int\dots\int\exp\left\{-N^2\left [\frac 1N\sum^N_{i =1}v(\lambda_j;t_1,\dots, t_\nu)-\frac{1}{N^2}\sum_{j\neq\ell}\log |\lambda_j-\lambda_l|\right ] \right\}\,d^N\lambda, \] where \[ V(\lambda;t_1,\dots, t_\nu)\overset{\text{def}}=\tfrac 12\lambda^2+ \sum^\nu_{k=1} t_k\lambda^k, \] and \(t_1,\dots,t_\nu\) are parameters.
Theorem 1.1 is related to the expansion \(\log\left(\frac{Z_N(t)}{Z_N(0)}\right)=N^2e_0(t)+e_1 (t)+\frac {1}{N^2}e_2(t)+\dots\) for \(t\in\mathbb R^\nu\) with \(|t|\leq T\), \(t_\nu>\gamma \sum^{\nu-1}_{j=1}| t_j|\).
Theorem 1.3 is related to the calculation of \(e_g(t)\) via \(g\)-maps.
Theorem 1.4 is a technical tool relating \(Z_N(t)\) to statistical mechanics and orthogonal polynomials theory.

Reviewer: Nguyêñ Hô\=ng Thaí (Szczecin)

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80	Random linear operators
47A56	Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15B52	Random matrices (algebraic aspects)
30E25	Boundary value problems in the complex plane
33D45	Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

Keywords:

partition function; random matrix theory; large \(N\) expansion

Citations:

Zbl 0453.05035