Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

Given an n × n complex matrix A, letbe the empirical spectral distribution (ESD) of its eigenvalues λ i ∈ C, i = 1, . . . n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD µ 1 √ n An of a random matrix A n = (a ij ) 1≤i,j≤n where the random variables a ij − E(a ij ) are iid copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1

In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigner's semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent ␤, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains. E igenvalues of graphs are useful for controlling many graph properties and consequently have numerous algorithmic applications including low rank approximations, ‡ information retrieval (1), and computer vision. § Of particular interest is the study of eigenvalues for graphs with power-law degree distributions (i.e., the number of vertices of degree j is proportional to j Ϫ␤ for some exponent ␤). It has been observed by many research groups (2-8, ¶) that many realistic massive graphs including Internet graphs, telephone-call graphs, and various social and biological networks have power-law degree distributions.For the classical random graphs based on the Erdös-Rényi model, it has been proved by Füredi and Komlós that the spectrum of the adjacency matrix follows the Wigner semicircle law (9). Wigner's theorem (10) and its extensions have long been used for the stochastic treatment of complex quantum systems that lie beyond the reach of exact methods. The semicircle law has extensive applications in statistical and solid-state physics (21,22). Here we intend to reconcile these two schools of thought on eigenvalue distributions. To begin with, there are in fact several ways to associate a matrix to a graph. The usual adjacency matrix A associated with a (simple) graph has eigenvalues quite sensitive to the maximum degree (which is a local property). The combinatorial Laplacian D Ϫ A, with D denoting the diagonal degree matrix, is a major tool for enumerating spanning trees and has numerous applications (13, 14). Another matrix associated with a graph is the, which controls the expansion͞isoperimetrical properties (which are global) and essentially determines the mixing rate of a random walk on the graph. The traditional random matrices and random graphs are regular or almost regular, thus the spectra of all the above three matrices are basically the same (with possibly a scaling factor or a linear shift). However, for graphs with uneven degrees, the above three matrices can h...

Abstract. Let x be a complex random variable with mean zero and bounded variance σ 2 . Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ 1 , . . . , λn be the eigenvalues ofNn. Define the empirical spectral distribution µn of Nn by the formulaThe following well-known conjecture has been open since the 1950's:Circular law conjecture: µn converges to the uniform distribution µ∞ over the unit disk as n tends to infinity.We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices.The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.