A Riemann-Hilbert Approach to Asymptotic Problems Arising in the Theory of Random Matrix Models, and also in the Theory of Integrable Statistical Mechanics

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

“…This approach (compare [22,34]) uses the integrable form of the Fredholm operator (1.12), allowing us to connect the resolvent kernel to the solution of a Riemann-Hilbert problem. The latter can be analysed rigorously via the Deift-Zhou nonlinear steepest descent method.…”

“…Fredholm determinant in the right hand side of (1.7) admits the following asymptotic representation [26], ln det(I − K sin ) = − (πs) 2 2 − 1 4 ln(πs) + 1 12 ln 2 + 3ζ ′ (−1) + O s −1 , s → ∞, (1.8) where ζ ′ (z) is the derivative of the Riemann zeta-function (a rigorous proof for this expansion without the constant term was obtained independently by Widom and Suleimanov -see [22] for more historical details; a rigorous proof including the constant terms was obtained independently in [27,41] -see also [18]). This remarkable formula yields one of the most important results in random matrix theory, i.e.…”

“…Now that we have defined the integral kernel (1.11) let us connect it to a Riemann-Hilbert problem: The given kernel belongs to an algebra of integrable operators first introduced in [34], see also [22]: Let Σ be an oriented contour in the complex plane C such as a Jordan curve. We are interested in operators of the form λI + K on L 2 (Σ), where K denotes an integral operator with kernel…”

“…[22]) to use the Bessel functions H (1) 0 (ζ) and H (2) 0 (ζ) for our construction. The latter Hankel functions of first and second kind are unique independent solutions to Bessel's equation…”

Integrable operators and canonical differential systems

“…has important applications in the theory of the random matrices(see [4], [7], [8], [14]). The general case was used in the spectral theory of the non-selfadjoint operators [12], [13].…”