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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.

Mathematics. --- Potential theory (Mathematics). --- System theory. --- Probabilities. --- Mathematical physics. --- Physics. --- Mathematical Physics. --- Probability Theory and Stochastic Processes. --- Potential Theory. --- Complex Systems. --- Mathematical Methods in Physics. --- Statistical Physics and Dynamical Systems. --- Math --- Science --- Distribution (Probability theory. --- Statistical physics. --- Physics --- Mathematical statistics --- Physical mathematics --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistical methods --- Mathematics --- Dynamical systems. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Risk --- Potential theory (Mathematics) --- Dynamics.