Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

/pdf/combinatorial-stochastic-processes-4nf5lxpd0t.pdf

Combinatorial Stochastic Processes

From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

/pdf/introduction-to-random-graphs-3cfvwg09nu.pdf

Introduction to random graphs

The elements of many classical combinatorial structures can be naturally decomposed into components Permutations can be decomposed into cycles, polynomials over a finite field into irreducible factors, mappings into connected components In all of these examples, and in many more, there are strong similarities between the numbers of components of different sizes that are found in the decompositions of `typical' elements of large size For instance, the total number of components grows logarithmically with the size of the element, and the size of the largest component is an appreciable fraction of the whole This book explains the similarities in asymptotic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition The discussion is conducted in the language of probability, enabling the theory to be developed under rather general and explicit conditions; for the finer conclusions, Stein's method emerges as the key ingredient The book is thus of particular interest to graduate students and researchers in both combinatorics and probability theory

Logarithmic Combinatorial Structures: A Probabilistic Approach

There are infinitely many Carmichael numbers

Let c be a positive constant. Suppose that r = o(n5/12) and the members of $({{[n]}\atop r})$ are chosen sequentially at random to form an intersecting hypergraph ${\cal H}$. We show that whp (A sequence of events ${\cal E}_1,\ldots,{\cal E}_n,\ldots$ is said to occur with high probability (whp) $\lim_{n\to\infty}\Pr({\cal E}_n)=1$.) ${\cal H}$ consists of a simple hypergraph ${\cal S}$ of size ˜(r/n1/3), a distinguished vertex v and all r-sets that contain v and meet every edge of ${\cal S}$. This is a continuation of the study of such random intersecting systems started in (Bohman et al., Electronic J Combinatorics (2003) R29) where the case r = O(n1/3) was considered. To obtain the stated result we continue to investigate this question in the range E(n1/3) d r d o(n5/12). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

pdf/randomly-generated-intersecting-hypergraphs-ii-haix9dh22w.pdf

Randomly generated intersecting hypergraphs II

It is known that the unit sphere, centered at the origin in ℝ
 n
 , has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝ
 n
 , and every ν > 0; there is a point r = (r
 1; r
 2;…;r
 n) such that:

 One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n;ℚ) is dense in O(n;ℝ) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n;ℂ) can likewise be approximated by matrices in U(n;ℚ(i))

/pdf/rational-points-on-the-unit-sphere-vdlgwwh7qx.pdf

Rational points on the unit sphere

https://www.math.drexel.edu/~eschmutz/PAPERS/recursive.pdf

Limit distribution for the maximum degree of a random recursive tree

The maximum degree in a random tree and related problems

If we compose sufficiently many random functions on a finite set, then the composite function will be constant. We determine the number of compositions that are needed, on average. Choose random functions $f_1, f_2,f_3,\dots $ independently and uniformly from among the $n^n$ functions from $[n]$ into $[n]$. For $t>1$, let $g_t=f_t\circ f_{t-1}\circ \cdots \circ f_1$ be the composition of the first $t$ functions. Let $T$ be the smallest $t$ for which $g_t$ is constant(i.e. $g_t(i)=g_t(j)$ for all $i,j$). We prove that $E(T)\sim 2n$ as $n\rightarrow\infty$, where $E(T)$ denotes the expected value of $T$.

/pdf/compositions-of-random-functions-on-a-finite-set-1amnxipu1k.pdf

Compositions of Random Functions on a Finite Set

For T E GL,(F,), let Q , ( T ) be the number of irreducible factors that the characteristic polynomial of T has. We prove that, for any fixed x , # { T : R,(T)<log n + x ) / # G L , , ( F , ) + ( l / G ) j tX e-rz’2 as n + m .

A central limit theorem on gln(fq)

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