Description
Book SynopsisThe goal of this book is to analyse in detail 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. The book is self-contained, and develops enough of the theory from each area that a general reader can learn the subject directly from the text.
Trade ReviewThe 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." -
Zentralblatt Math"The book covers exciting results, and has a wealth of information." - Milós Bóna,
MAA Reviews"…[T]he book is carefully written and will serve as an excellent reference." - Terence Tao,
Mathematical ReviewsTable of Contents
- Introduction
- Poissonization and de-Poissonization
- Permutations and Young tableaux
- Bounds of the expected value of $\ell_N$
- Orthogonal polynomials, Riemann-Hilbert problems, and Toeplitz matrices
- Random matrix theory
- Toeplitz determinant formula
- Fredholm determinant formula
- Asymptotic results
- Schur measure and directed last passage percolation
- Determinantal point processes
- Tiling of the Aztec diamond
- The Dyson process and Brownian Dyson process
- Theory of trace class operators and Fredholm determinants
- Steepest-descent method for the asymptotic evaluation of integrals in the complex plane
- Basic results of stochastic calculus
- Bibliography
- Index
