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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of random variables ($X_n$)$ _{n\in{\mathbb{N}}}$. The random variables considered can be lattice or non-lattice distributed, and single or multi-dimensional; and one obtains precise estimates of the fluctuations $\mathbb{P}[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay log($\mathbb{P}[X_{n} \in t_{n}B]$In the special setting of mod-Gaussian convergence, we shall see that our approach allows us to identify the scale at which the central limit theorem ceases to hold and we are able to quantify the "breaking of symmetry" at this critical scale thanks to the residue or limiting function occurring in mod-f convergence. In particular this provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory (multi-dimensional random walks, random point processes), number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd˝os-Renyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of “weakly dependent” random variables. Although the latter methods can only be applied in the more restrictive setting of mod-Gaussian convergence, the large number as well as the variety of examples which are covered there hint at a universality class for second order fluctuations.

Mod-ϕ Convergence: Normality Zones and Precise Deviations

In this paper, we are interested in the asymptotic size of the rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order n, so it does not fit in the context of the work of Biane (Int Math Res Notices 4:179–192, 2001) and Śniady (Probab. Theory Relat Fields 136:263–297, 2006). Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method also works for other measures, for example those coming from Schur–Weyl representations.

Asymptotics of q-plancherel measures

In this paper, we prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: 

1.

the classical simple compact Lie groups: special orthogonal groups SO(n), special unitary groups SU(n) and compact symplectic groups USp(n);




2.

the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces when q = 1): SO(p + q)/(SO(p)×SO(q)), SU(p + q)/S(U(p)×U(q)) and USp(p + q)/(USp(p)×USp(q));




3.

the spaces of real, complex and quaternionic structures: SU(n)/SO(n), SO(2n)/ U(n), SU(2n)/USp(n) and USp(n)/UU(n).






Denoting μt the law of the Brownian motion at time t, we give explicit lower bounds for dTV(μt,Haar) if $t t_{\text{cut-of\/f}}$. This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in $\mathfrak{S}_{n}$.

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The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

We show that Kerov's central limit theorem related to the fluctuations of Young diagrams under the Plancherel measure extends to the case of Schur-Weyl measures, which are the probability measures on partitions associated to the representations of the symmetric groups $S_n$ on tensor products of vector spaces $V^{\otimes n}$ (cf. arXiv:math/0006111). More precisely, the fluctuations are exactly the same up to a translation of the diagrams along the x-axis. Our proof is inspired by the one given by Ivanov and Olshanski in arXiv:math/0304010 for the Plancherel measure, and it relies on the combinatorics of the algebra of observable of diagrams. We also use Sniady's theory of cumulants of observables, cf. arXiv:math/0501112.

Kerov's central limit theorem for Schur-Weyl measures of parameter 1/2

Representation Theory of Symmetric Groups is the most up-to-date abstract algebra book on the subject of symmetric groups and representation theory Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint 
This book is an excellent way of introducing today’s students to representation theory of the symmetric groups, namely classical theory From there, the book explains how the theory can be extended to other related combinatorial algebras like the Iwahori-Hecke algebra 
In a clear and concise manner, the author presents the case that most calculations on symmetric group can be performed by utilizing appropriate algebras of functions Thus, the book explains how some Hopf algebras (symmetric functions and generalizations) can be used to encode most of the combinatorial properties of the representations of symmetric groups 
Overall, the book is an innovative introduction to representation theory of symmetric groups for graduate students and researchers seeking new ways of thought

Pierre-Loïc Méliot

Papers

Mod-ϕ Convergence: Normality Zones and Precise Deviations

Asymptotics of q-plancherel measures

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Kerov's central limit theorem for Schur-Weyl measures of parameter 1/2

Representation Theory of Symmetric Groups