This book elaborates on the asymptotic behaviour, when N is large, of
certain N-dimensional integrals which typically occur in random
matrices, or in 1+1 dimensional quantum integrable models solvable by
the quantum separation of variables. The introduction presents the
underpinning motivations for this problem, a historical overview, and a
summary of the strategy, which is applicable in greater generality. The
core aims at proving an expansion up to o(1) for the logarithm of the
partition function of the sinh-model. This is achieved by a combination
of potential theory and large deviation theory so as to grasp the
leading asymptotics described by an equilibrium measure, the
Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse
the equilibrium measure, the Schwinger-Dyson equations and the boostrap
method to finally obtain an expansion of correlation functions and the
one of the partition function. This book is addressed to researchers
working in random matrices, statistical physics or integrable systems,
or interested in recent developments of asymptotic analysis in those
fields.