arXiv subjects

Erik Thörnblad

Publications and source records attributed to Erik Thörnblad.

10 recordsLinked to original sources

arXiv2019

Existence of symmetric maximal noncrossing collections of $k$-element sets

We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of $\left\{ 1, \dots, n \right\}$ that are invariant under adding $k\pmod n$ to all indices. Our main result is that such a collection exists if and only if $k$ is congruent to $0, 1$ or $-1$ modulo $n/\operatorname{GCD}(k,n)$. Moreover, we present some algebraic consequences of our result related to self-injective Jacobian algebras.

math.CO↗

arXiv2017

Counting Quasi-Idempotent Irreducible Integral Matrices

Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences and their asymptotics. Such matrices turn out to be related to generalized compositions and generalized partitions.

math.CO↗

arXiv2016

The greedy walk on an inhomogeneous Poisson process

The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. Our primary interest is whether the walk visits all points of the point process, and we determine sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.

math.PR↗

arXiv2016

Tournament limits: Degree distributions, score functions and self-converseness

Motivated by known results for finite tournaments, we define and study the score functions of tournament kernels and the degree distributions of tournament limits. Our main theorem completely characterises those distributions that appear as the degree distribution of some tournament limit and those functions that appear as the score function of some tournament kernel. We also show that only the uniform distribution can be realised as the outdegree distribution of a unique tournament limit. Finally we define self-converse tournament limits and kernels and characterise their degree distributions and score functions.

math.CO↗

arXiv2016

Another proof of Moon's theorem on generalised tournament score sequences

Landau \cite{Landau1953} showed that a sequence $(d_i)_{i=1}^n$ of integers is the score sequence of some tournament if and only if $\sum_{i\in J}d_i \geq \binom{|J|}{2}$ for all $J\subseteq \{1,2,\dots, n\}$, with equality if $|J|=n$. Moon \cite{Moon63} extended this result to generalised tournaments. We show how Moon's result can be derived from Landau's result.

math.CO↗

arXiv2016

Eplett's theorem for self-converse generalised tournaments

The converse of a tournament is obtained by reversing all arcs. If a tournament is isomorphic to its converse, it is called self--converse. Eplett provided a necessary and sufficient condition for a sequence of integers to be realisable as the score sequence of a self--converse tournament. In this paper we extend this result to generalised tournaments.

math.CO↗

arXiv2016

Decomposition of tournament limits

The theory of tournament limits and tournament kernels (often called graphons) is developed by extending common notions for finite tournaments to this setting; in particular we study transitivity and irreducibility of limits and kernels. We prove that each tournament kernel and each tournament limit can be decomposed into a direct sum of irreducible components, with transitive components interlaced. We also show that this decomposition is essentially unique.

math.CO↗

arXiv2015

Almost sure convergence of vertex degree densities in the vertex-splitting model

We study the limiting degree distribution of the vertex splitting model introduced in \cite{DDJS:2009}. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.

math.PR↗

arXiv2015

The dominating colour of an infinite Pólya urn model

We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with another ball of the same colour. With probability 1-p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri, and Thörnblad. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1-p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p=1).

math.PR↗

arXiv2014

Asymptotic degree distribution of a duplication-deletion random graph model

We study a discrete-time duplication-deletion random graph model and analyse its asymptotic degree distribution. The random graphs consists of disjoint cliques. In each time step either a new vertex is brought in with probability $0<p<1$ and attached to an existing clique, chosen with probability proportional to the clique size, or all the edges of a random vertex are deleted with probability $1-p$. We prove almost sure convergence of the asymptotic degree distribution and find its exact values in terms of a hypergeometric integral, expressed in terms of the parameter $p$. In the regime $0<p<\frac{1}{2}$ we show that the degree sequence decays exponentially at rate $\frac{p}{1-p}$, whereas it satisfies a power-law with exponent $\frac{p}{2p-1}$ if $\frac{1}{2}<p<1$. At the threshold $p=\frac{1}{2}$ the degree sequence lies between a power-law and exponential decay.

math.PR↗