Roots of polynomials of degrees 3 and 4
We present the solutions of equations of degrees 3 and 4 using Galois theory and some simple Fourier analysis for finite groups, together with historical comments on these and other solution methods.
math.HO↗
arXiv subjects
Svante Janson
Publications and source records attributed to Svante Janson.
On covering by translates of a set
In this paper we study the minimal number of translates of an arbitrary subset $S$ of a group $G$ needed to cover the group, and related notions of the efficiency of such coverings. We focus mainly on finite subsets in discrete groups, reviewing the classical results in this area, and generalizing them to a much broader context. For example, we show that while the worst-case efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and $n$ large, almost every $k$-subset of any given $n$-element group covers $G$ with close to optimal efficiency.
math.CO↗
Correlations for paths in random orientations of G(n,p) and G(n,m)
We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combined probability space, of the events a -> s and s -> b. For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p p_c the correlation is positive for large enough n. We conjecture that for a fixed n\ge 27 the correlation changes sign three times for three critical values of p. For G(n,m) it is similarly proved that, with $p=m/\binom{n}{2}$, there is a critical p_c that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any k directed edges in G(n,m), is thought to be of independent interest. We present exact recursions to compute P(a -> s)$ and P(a -> s, s -> b)$. We also briefly discuss the corresponding question in the quenched version of the problem.
math.PR↗
Phase transitions for modified Erdös-Rényi processes
A fundamental and very well studied region of the Erdös-Rényi process is the phase transition at n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman-Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.
math.CO↗
Hitting times for random walks with restarts
The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In particular, we show that in the planar square lattice, $T(x,0)$ is asymptotic to $2|x|^2\log|x|$ as $|x| \to \infty$. The proof hinges on the local variance of the potential kernel $h$ being almost constant on the level sets of $h$. We also show how the same method yields precise second order asymptotics for hitting times of a random walk (without restarts) in a lattice disk.
math.PR↗
Absolutely Continuous Compensators
We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Cinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and space, have such a representation.
math.PR↗
Moments of Gamma type and the Brownian supremum process area
We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occuring in the study of hashing with linear displacement, and the general results are applied to these variables.
math.PR↗
The cut metric, random graphs, and branching processes
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model we introduced recently, as well as related results of Bollobás, Borgs, Chayes and Riordan, all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering.
math.PR↗
The maximum of Brownian motion with parabolic drift
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
math.PR↗
Renewal theory in analysis of tries and strings
We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for b-tries and Patricia tries; Khodak and Tunstall codes.
cs.DS↗
On the number of perfect matchings in random lifts
Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let X_G be the number of perfect matchings in a random lift of G. We study the distribution of X_G in the limit as n tends to infinity, using the small subgraph conditioning method. We present several results including an asymptotic formula for the expectation of X_G when G is d-regular, d\geq 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of X_G, with full details given for two example multigraphs, including the complete graph K_4. To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.
math.CO↗
Susceptibility of random graphs with given vertex degrees
We study the susceptibility, i.e., the mean cluster size, in random graphs with given vertex degrees. We show, under weak assumptions, that the susceptibility converges to the expected cluster size in the corresponding branching process. In the supercritical case, a corresponding result holds for the modified susceptibility ignoring the giant component and the expected size of a finite cluster in the branching process; this is proved using a duality theorem. The critical behaviour is studied. Examples are given where the critical exponents differ on the subcritical and supercritical sides.
math.CO↗
Sparse random graphs with clustering
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.
math.PR↗
Threshold graph limits and random threshold graphs
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
math.CO↗
Zeros of Sections of the Binomial Expansion
We examine the asymptotic behaviour of the zeros of sections of the binomial expansion. That is, we consider the distribution of zeros of $\displaystyle B_{r,n}(z) = \sum_{k=0}^r {n \choose k} z^k$, where $1 \le r < n$.
math.CV↗
The Mahonian probability distribution on words is asymptotically normal
The Mahonian statistic is the number of inversions in a permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function for this statistic is the $q$ analog of the multinomial coefficient $\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating function is the normalization of the latter. We give two proofs that the distribution is asymptotically normal. The first is {\it computer-assisted}, based on the method of moments. The Maple package {\tt MahonianStat}, available from the webpage of this article, can be used by the reader to perform experiments and calculations. Our second proof uses characteristic functions. We then take up the study of a local limit theorem to accompany our central limit theorem. Here our result is less general, and we must be content with a conjecture about further work. Our local limit theorem permits us to conclude that the coeffiecients of the $q$-multinomial are log-concave, provided one stays near the center (where the largest coefficients reside.)
math.CO↗
A functional combinatorial central limit theorem
The paper establishes a functional version of the Hoeffding combinatorial central limit theorem. First, a pre-limiting Gaussian process approximation is defined, and is shown to be at a distance of the order of the Lyapounov ratio from the original random process. Distance is measured by comparison of expectations of smooth functionals of the processes, and the argument is by way of Stein's method. The pre-limiting process is then shown, under weak conditions, to converge to a Gaussian limit process. The theorem is used to describe the shape of random permutation tableaux.
math.PR↗
Long and short paths in uniform random recursive dags
In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant σsuch that S_n/log(n) tends to σin probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to σin probability.
math.PR↗