## A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics.(English)Zbl 0936.47028

Random matrix theory was introduced for the needs of theoretical physics, in particular, in Wigner’s investigations on nuclear physics. Now this theory is used in statistical mechanics, field theory, number theory and many other areas of applied mathematics.
The problem of the paper under review is the investigation of the asymptotic behavior as $$x\to\infty$$ of the Fredholm determinant $$P_x= \det(1-K_x)$$ where $$K_x$$ is the (trace class) operator with kernel $$K_x(z,z')=\frac{\sin x(z-z')}{\pi(z-z')}$$ acting on $$L^2(J,dz)$$, $$J=\sum_{k=0} ^n (a_k,b_k)$$ is a union of $$n+1$$ disjoint intervals in $$\mathbb R$$. The determinant $$P_x$$ is the probability of finding no eigenvalues in the union of intervals $$\frac{xJ}{\pi}=\sum_{k=0} ^n(\frac{xa_k}{\pi},\frac{xb_k}{\pi})$$ for a random Hermitian matrix chosen from the Gaussian unitary ensemble.
After a short résumé of the known results (of H. Widom and others) in this field based on the Jacobi inversion problem and on a development of the Szegő method the authors give an other idea for the solution of the problem connected with the Riemann-Hilbert problem on a Riemann surface of a hyperelliptic curve. We give the statement of the main theorem.
Let $v=\left(\begin{matrix} 0 & 1\\ -1 & 2 \end{matrix} \right) \qquad \text{and}\qquad v_x(z)=e^{izx\sigma_3} v e^{-izx\sigma_3}= \left(\begin{matrix} 0 & e^{izx}\\ e^{-izx} & 2 \end{matrix} \right)$ where $$\sigma_3$$ is the Pauli matrix $\left(\begin{matrix} 1 & 0\\ 0 & -1 \end{matrix} \right).$ Let $$m(z)$$ be an analytic $$2 \times 2$$-matrix function in $$\mathbb C\setminus \overline J$$ with $$L^2(J,dz)$$-boundary values $$m_{\pm}$$ on $$J$$.
Theorem. For each fixed $$x\geq 0$$ the $$2 \times 2$$-matrix valued function $$m(z)=m(z;x)$$ solve the RH problem $\begin{cases} m(z), &\text{is analytic in $$\mathbb C \setminus \overline J$$},\\ m_{+}=m_{-}(z) v_x(z), &z \in J,\\ m(z) \to I &\text{as $$z \to \infty$$}.\end{cases}$ Then $\frac{d}{dx}\log P_x=\frac{d}{dx}\log \det (1-K_x)=i(((m_1(x))_{22}-(m_1(x))_{11}),$ where $m(z)=I+\frac{m_1(x)}{z}+O\biggl(\frac{1}{z^2}\biggr), \quad \text{as $$z\to \infty$$}.$ The second theorem asserts that asymptotics of the $$\frac{d}{dx}\log P_x$$ can be given in terms of $$\theta$$ functions on the hyperelliptic curve $$y^2= \prod_{i=0} ^{n}[(z-a_i)(z-b_i)]$$. The complete asymptotic expansions are given as well as some remarks on the methods and connections with other fields. The authors use a very rich set of methods from Fourier analysis and the deepest descent method to Diophantine approximation and Painlevé transcendents.
Finally, an application of the results to the investigation of the isotropic $$\text{spin-}\frac{1}{2}XY$$ model in a moderate transverse magnetic field is given.

### MSC:

 47G10 Integral operators 15B52 Random matrices (algebraic aspects) 30E25 Boundary value problems in the complex plane 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 82B23 Exactly solvable models; Bethe ansatz
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