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.

/pdf/stochastic-geometry-and-random-graphs-for-the-analysis-and-1e671n4af3.pdf

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.

/pdf/an-introduction-to-random-matrices-1gerxro6r6.pdf

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

Palm measures for Dirac operators and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si2.svg"><mml:msub><mml:mrow><mml:mo class="qopname">Sine</mml:mo></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:math> process

We construct a random Schrodinger operator on a subset of the hexagonal lattice and study its smallest positive eigenvalues. Using an asymptotic mapping, we relate them to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the random Schrodinger operator. We also relate the distribution of its k smallest positive eigenvalues to the nonintersecting partition functions of order k.

/pdf/tracy-widom-fluctuations-in-2d-random-schrodinger-operators-4uhr5m51bk.pdf

Tracy-Widom fluctuations in 2D random Schrodinger operators

We give lower bounds for the electrical resistance between vertices in the Schreier graphs of the action of the linear (degree 1) and quadratic (degree 2) mother groups on the orbit of the zero ray. These bounds, combined with results of \cite{JNS} show that every quadratic activity automaton group is amenable. The resistance bounds use an apparently new "weighted" version of the Nash-Williams criterion which may be of independent interest.

Bálint Virág

Papers

Palm measures for Dirac operators and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e20" altimg="si2.svg"><mml:msub><mml:mrow><mml:mo class="qopname">Sine</mml:mo></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:math> process

Tracy-Widom fluctuations in 2D random Schrodinger operators

Amenability of quadratic automaton groups