Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth.

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Random Matrix Theory and Wireless Communications

Let x(1) denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x(1) is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a p­variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painleve II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.

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On the distribution of the largest eigenvalue in principal components analysis

We study $ \mathcal{N} = 2 $ supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.

http://math.univ-lyon1.fr/~attal/Okounkov/okounkov-nekrasov

Seiberg-Witten theory and random partitions

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.

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An Introduction to Random Matrices

Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution,

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On the distribution of the length of the longest increasing subsequence of random permutations

We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy–Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).

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Shape fluctuations and random matrices

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel [Ge] for the Poissonization of the distribution function of $l_N$.

On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

1,156,645. Ornamenting. E. I. DU PONT DE NEMOURS & CO. 14 June, 1967 [15 June, 1966], No. 27377/67. Heading B6G. [Also in Divisions B5 and C3] Decorative articles are obtained by printing and laminating surface-receptive polyvinyl fluoride film with an adhesive/printing ink vehicle composition comprising an organic solvent solution of (1) a vinyl polymer having attached to the main chain thereof monovalent radicals of the formula wherein Z is a tetravalent aliphatic hydrocarbon radical of 2 or 3 carbon atoms, R is H or C 1 -C 8 alkyl, R 1 is H, OH or C 1 -C 8 alkyl and R 11 is H or C 1 -C 8 non-tertiary alkyl or C 1 -C 8 #- hydroxyalkyl, the amino nitrogen content of said vinyl polymer being at least 0A01%, based on the weight of the vinyl polymer, and (2) an epoxy ether or epoxy thioether reaction product characterized in that the precursor epoxy ether (or thioether) is partially reacted with an ethylenically unsaturated compound of the formula wherein R 3 is H or carboxyl or C 1 -C 4 alkyl, R 4 is an alkyleneoxy, alkyleneoxy carbonyl, divalent aromatic, other divalent aliphatic or divalent alicyclic radical, or when A is R 4 may be a single bond, A is -O-, -TH-, and R 7 is H or a carboxy, carboalkoxy, hydroxy, sulphhydryl, sulphonyl, amino, non-hindering alkyl or non-hindering alkyl-aryl group, said reaction product having unhindered ethylenic unsaturation, an epoxide equivalent of from 50 to 3000, and a bromine number of from 0A1 to 100; said components (1) and (2) being present in a ratio, by weight, of from 1 : 99 to 99 : 1. Preferred vinyl polymers contain 80-97% by weight of units of (meth)-acrylate esters of C 1 - C 8 aliphatic alcohols, and have the amino-ester groups present in amino-ethyl, amino-propyl or amino-hydroxypropyl methacrylate units, the amino group generally being introduced by modification of a suitable preformed polymer (e.g. by imination of a polymer containing carboxylic acid groups). The epoxy-(thio)- ether reaction product is generally derived from an epoxy compound containing on average more than one epoxy group per molecule; the preferred precursor epoxy-compounds being diphenylol-propane diglycidyl ether resins (and corresponding thio - derivatives). The other component of the epoxy reaction product is preferably an olefinically unsaturated carboxylic acid. The polyvinyl fluoride film is rendered surface receptive by chemical, electrical or flame-treatment methods. It is then printed with the aforesaid solution and laminated to a substrate composed of a thermosetting resin, optionally reinforced with, e.g. glass fibres. The laminating/printing solution may contain a pigment, e.g. a brown pigment or titanium dioxide. In the examples, the polyvinyl fluoride film is printed with the solution in a pattern by stamping a series of \" X's \" into a template, spraying the solution over the X's and drying; after laminating with a polyester/ styrene/glass fibre mat there are obtained exterior trim panels suitable for use in station wagons.

Kurt Johansson

Papers

On the distribution of the length of the longest increasing subsequence of random permutations

Shape fluctuations and random matrices

Shape Fluctuations and Random Matrices

On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

On fluctuations of eigenvalues of random Hermitian matrices