Combinatorics and Random Matrix Theory (Graduate Studies in Mathematics)

Over the last fifteen years a variety of problems in combinatorics has been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a "stochastic special function theory" for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail. Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

• Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach (Courant Lecture Notes)
• The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition (Springer Series in Statistics)
• Probability on Trees and Networks (Cambridge Series in Statistical and Probabilistic Mathematics)
• Stochastic Equations in Infinite Dimensions (Encyclopedia of Mathematics and its Applications)
• Deep Learning (Adaptive Computation and Machine Learning series)
• Pattern Recognition and Machine Learning (Information Science and Statistics)
• Random Matrix Theory: Invariant Ensembles and Universality (Courant Lecture Notes) (Courant Lecture Notes in Mathematics)
• Category Theory in Context (Aurora: Dover Modern Math Originals)
• Eigenvalue Distribution of Large Random Matrices (Mathematical Surveys and Monographs)
• Machine Learning: A Probabilistic Perspective (Adaptive Computation and Machine Learning series)