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Algebra --- Functional analysis --- Operational research. Game theory --- Discrete mathematics --- algebra --- matrices --- discrete wiskunde --- stochastische analyse --- functies (wiskunde) --- kansrekening

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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

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

Random matrices. --- Matrices. --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices, Random --- Matrices

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Random matrix theory has developed in the last few years, in connection with various fields of mathematics and physics. These notes emphasize the relation with the problem of enumerating complicated graphs, and the related large deviations questions. Such questions are also closely related with the asymptotic distribution of matrices, which is naturally defined in the context of free probability and operator algebra. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006. Lectures were also given by Maury Bramson and Steffen Lauritzen.

Mathematics. --- Matrix theory. --- Functional analysis. --- Combinatorics. --- Distribution (Probability theory). --- Functional Analysis. --- Probability Theory and Stochastic Processes. --- Linear and Multilinear Algebras, Matrix Theory. --- Random matrices --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Asymptotic expansions --- Asymptotic developments --- Matrices, Random --- Algebra. --- Probabilities. --- Discrete mathematics. --- Discrete Mathematics. --- Combinatorics --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Matrices --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical