Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to graphical enumeration

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.Comment: 44 pages, 4 figures. To appear, International Mathematics Research Notice

Correlation Functions for a Chain of Short Range Oscillators

We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t-13 for position and momentum correlations and as t-23 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t-14 for position and momentum correlators and with rate t-12 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly