Proceedings of Symposia in Applied Mathematics 1991; 196 pp; hardcover Volume: 44 ISBN10: 082185500X ISBN13: 9780821855003 List Price: US$54 Member Price: US$43.20 Order Code: PSAPM/44
 Probabilistic methods have become a vital tool in the arsenal of every combinatorialist. The theory of random graphs is still a prime area for the use of probabilistic methods, and, over the years, these methods have also proved of paramount importance in many associated areas such as the design and analysis of computer algorithms. In recent years, probabilistic combinatorics has undergone revolutionary changes as the result of the appearance of some exciting new techniques such as martingale inequalities, discrete isoperimetric inequalities, Fourier analysis on groups, eigenvalue techniques, branching processes, and rapidly mixing Markov chains. The aim of this volume is to review briefly the classical results in the theory of random graphs and to present several of the important recent developments in probabilistic combinatorics, together with some applications. The first paper contains a brief introduction to the theory of random graphs. The second paper reviews explicit constructions of randomlike graphs and discusses graphs having a variety of useful properties. Isoperimetric inequalities, of paramount importance in probabilistic combinatorics, are covered in the third paper. The chromatic number of random graphs is presented in the fourth paper, together with a beautiful inequality due to Janson and the important and powerful SteinChen method for Poisson approximation. The aim of the fifth paper is to present a number of powerful new methods for proving that a Markov chain is "rapidly mixing" and to survey various related questions, while the sixth paper looks at the same topic in a very different context. For the random walk on the cube, the convergence to the stable distribution is best analyzed through Fourier analysis; the final paper examines this topic and proceeds to several more sophisticated applications. Open problems can be found throughout each paper. Table of Contents  B. Bollobás  Random graphs
 F. K. Chung  Constructing randomlike graphs
 I. Leader  Discrete isoperimetric inequalities
 B. Bollobás  Random graphs revisited
 U. Vazirani  Rapidly mixing Markov chains
 M. Dyer and A. Frieze  Computing the volume of convex bodies: A case where randomness provably helps
 P. Diaconis  Finite Fourier methods: Access to tools
