We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.

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Stochastic geometry and random graphs for the analysis and design of wireless networks

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

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

La loi de Tracy–Widom $\beta$ est la limite de la plus grande valeur propre des ensembles $\beta$ de matrices aleatoires lorsque leur taille tend vers l’infini. Nous utilisons la representation par l’operateur stochastique d’Airy pour montrer que lorsque $a\to\infty$ la queue de la loi de Tracy–Widom verifie : \[P(\mathit{TW}_{\beta}>a)=a^{-(3/4)\beta+\mathrm{o}(1)}\exp\biggl(-\frac{2}{3}\beta a^{3/2}\biggr).\]

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The right tail exponent of the Tracy–Widom $\beta$ distribution

We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. 
We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. 
Kesten showed that if a Cayley graph has the same spectral radius as its universal cover, then it must be a tree. We generalize this to unimodular random graphs.

The measurable Kesten theorem

We show that in the point process limit of the bulk eigenvalues of β-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size λ is given by (κβ+o(1))λγβ exp((−β/64)λ2+(β/8−1/4)λ) as λ→∞, where γβ=1/4(β/2+2/β−3) and κβ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157–165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron–Martin–Girsanov transformation in stochastic calculus.

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Large gaps between random eigenvalues.

This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric (Cheeger) constant at least i. We prove a conjecture by Benjamini, Lyons and Schramm (1999) that in such graphs the random walk escapes with a positive lim inf speed. We also show that anchored expansion implies a heat-kernel decay bound of order exp(—cn 1/3).

Anchored expansion and random walk

Let G be a d-regular graph of sufficiently large-girth depending on parameters k and r and µ be a random process on the vertices of G produced by a randomized local algorithm of radius r. We prove the upper bound k+1-2k/d1d-1k for the absolute value of the correlation of values on pairs of vertices of distance k and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the d-regular tree. In that case we give an explicit description for the closure of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infinite d-regular tree has spectral radius 2d-1. Graphings with this spectral gap are infinite analogues of finite Ramanujan graphs and they are interesting on their own right. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 424-435, 2015

Bálint Virág

Papers

The right tail exponent of the Tracy–Widom $\beta$ distribution

The measurable Kesten theorem

Large gaps between random eigenvalues.

Anchored expansion and random walk

Ramanujan graphings and correlation decay in local algorithms