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

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

This important text and reference for researchers and students in machine learning, game theory, statistics and information theory offers a comprehensive treatment of the problem of predicting individual sequences. Unlike standard statistical approaches to forecasting, prediction of individual sequences does not impose any probabilistic assumption on the data-generating mechanism. Yet, prediction algorithms can be constructed that work well for all possible sequences, in the sense that their performance is always nearly as good as the best forecasting strategy in a given reference class. The central theme is the model of prediction using expert advice, a general framework within which many related problems can be cast and discussed. Repeated game playing, adaptive data compression, sequential investment in the stock market, sequential pattern analysis, and several other problems are viewed as instances of the experts' framework and analyzed from a common nonstochastic standpoint that often reveals new and intriguing connections.

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Prediction, learning, and games

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

q‐binary search trees are obtained from words, equipped with the geometric distribution instead of permutations. The average and variance of the height are computed, based on random words of length n, as well as a Gaussian limit law.

/pdf/the-height-of-q-binary-search-trees-1lnqr2uhz1.pdf

The Height of q-Binary Search Trees

We study the number of divisors in residue classes modulo m and prove, for example, that the exact equidistribution holds for almost all natural numbers coprime to m in the sense of natural density if and only if m = 2kp1p2…ps, where k and s are non-negative integers and pj are distinct Fermat primes. We also provide a general and exact lower bound for the proportion of divisors in the residue class 1 mod m. The same combinatorial technique using Davenport's constant leads to exact lower bounds for the number of representations of a natural number by a given binary quadratic form with a negative discriminant.

Equidistribution of divisors in residue classes and representations by binary quadratic forms

It is shown that the number of leftist trees of size n in a simply generated family of trees is asymptotically given by \sim α c

 n
 n

 -3/2
 with certain constants α>0 and c >1 . Furthermore, it is proved that the number of leaves in leftist trees with n nodes satisfies a central limit theorem.

The Asymptotic Number of Leftist Trees

We consider a random k-tree Gn,k that is uniformly selected from the class of labelled k-trees with n+ k vertices. Since 1-trees are just trees, it is well-known that Gn,1 (after scaling the distances by 1/(2 √ n)) converges to the Continuum Random Tree Te. Our main result is that for k 6= 1, the random k-tree Gn,k, scaled by (kHk−1 + 1)/(2 √ n) where Hk−1 is the (k − 1)-th Harmonic number, converges to the Continuum Random Tree Te, too. In particular this shows that the diameter as well as the typical distance of two vertices in a random k-tree Gn,k are of order √ n.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611974324.7

Scaling limit of random k -trees.

It is proved that for almost all s × s-matrices A with largest eigenvalue λ(A)>1 the sequence of powers (Ap(n)) n=1 ∞ is completely uniformly distributed modulo 1, where p(n) are different positive integers. Furthermore a constructive example for a matrix A is given such that the sequence (An) n=1 ∞ is completely uniformly distributed.

Michael Drmota

Papers

The Height of q-Binary Search Trees

Equidistribution of divisors in residue classes and representations by binary quadratic forms

The Asymptotic Number of Leftist Trees

Scaling limit of random k -trees.

Completely uniformly distributed sequences of matrices