arXiv subjects

Fabian Burghart

Publications and source records attributed to Fabian Burghart.

11 recordsLinked to original sources

arXiv2026

On the intersection of pairs of trees

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson distribution with expected value $2$. This is applied to show a Poisson limit law for the number of common edges in two independent random spanning trees of an Erdős--Rényi random graph $G(n,p)$ for constant~$p$, as well as a central limit theorem in the case where $p\to 0$ and $p\geq n^{-2/3+\varepsilon}$. We also use the same method to prove an analogous result for complete multipartite graphs.

math.CO↗

arXiv2026

Ancestries in random $d$-DAGs

We consider a random recursive DAG $G_n$ on the vertex set $[n]$ where every vertex $i\geq 2$ has out-degree $d$, with the targets chosen uniformly at random among the earlier $i-1$ vertices. For this model, we propose a novel way to investigate the descendants of $n$ (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process $a_i(n)$ of a vertex $i$ is defined as the number of ancestors of $i$ in $G_n$, and is closely related to the evolutions of multi-draw Pólya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. This method yields the answer to two questions posed by Janson, the first on the size of the joint descendants of vertices $n$ and $n+1$, and the other on the location of the earliest non-descendant. We further prove limit theorems for the ancestry processes $a_i(n)$ depending on $i$, determine the location of the earliest source node, and provide an alternative proof of a first-moment result contained in Janson's work.

math.PR↗

arXiv2026

On Cycles in Multiset Permutations, Parking Functions, and Related Structures

In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.

math.CO↗

arXiv2025

Sharp Thresholds for Factors in Random Graphs

Let $F$ be a graph on $r$ vertices and let $G$ be a graph on $n$ vertices. Then an $F$-factor in $G$ is a subgraph of $G$ composed of $n/r$ vertex-disjoint copies of $F$, if $r$ divides $n$. In other words, an $F$-factor yields a partition of the $n$ vertices of $G$. The study of such $F$-factors in the Erdős-Rényi random graph dates back to Erdős himself. Decades later, in 2008, Johansson, Kahn and Vu established the thresholds for the existence of an $F$-factor for strictly 1-balanced $F$ -- up to the leading constant. The sharp thresholds, meaning the leading constants, were obtained only recently by Riordan and Heckel, but only for complete graphs $F=K_r$ and for so-called nice graphs. Their results rely on sophisticated couplings that utilize the recent, celebrated solution of Shamir's problem by Kahn. We extend the couplings by Riordan and Heckel to any strictly 1-balanced $F$ and thereby obtain the sharp threshold for the existence of an $F$-factor. In particular, we confirm the thirty year old conjecture by Rucínski that this sharp threshold indeed coincides with the sharp threshold for the disappearance of the last vertices which are not contained in a copy of $F$.

math.CO↗

arXiv2025

The Einstein Relation on Metric Measure Spaces

This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at Hölder regular transformations and how they influence the local walk dimension and describe the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.

math.FA↗

arXiv2024

Concatenating Random Matchings

We consider the concatenation of $t$ uniformly random perfect matchings on $2n$ vertices, where the operation of concatenation is inspired by the multiplication of generators of the Brauer algebra $\mathfrak{B}_n(δ)$. For the resulting random string diagram $\mathsf{Br}_n(t)$, we observe a giant component if and only if $n$ is odd, and as $t\to\infty$ we obtain asymptotic results concerning the number of loops, the size of the giant component, and the number of loops of a given shape. Moreover, we give a local description of the giant component. These results mainly rely on the use of renewal theory and the coding of connected components of $\mathsf{Br}_n(t)$ by random vertex-exploration processes.

math.PR↗

arXiv2024

The hitting time of nice factors

Consider the random $u$-uniform hypergraph (or $u$-graph) process on $n$ vertices, where $n$ is divisible by $r>u\ge 2$. It was recently shown that with high probability, as soon as every vertex is covered by a copy of the complete $u$-graph $K_r$, it also contains a $K_r$-factor (RSA, Vol. 65 II, Sept. 2024). The hitting time result is obtained using a process coupling, which is based on the proof of the corresponding sharp threshold result (RSA, Vol. 61 IV, Dec. 2022). The latter, however, was not only derived for complete $u$-graphs, but for a broader class of so-called nice $u$-graphs. The purpose of this article is to extend the process coupling for complete $u$-graphs to the full scope of the sharp threshold result: nice $u$-graphs. As a byproduct, we obtain the extension of the hitting time result to nice $u$-graphs. Since the relevant combinatorial bounds in the proof for the $K_r$-case cannot be generalized, we introduce new arguments that do not only apply to nice u-graphs, but will be relevant for the broader class of strictly 1-balanced u-graphs. Further, we show how the remainder of the process coupling for the $K_r$-case can be utilized in a black-box manner for any u-graph. These advances pave the way for future generalizations.

math.CO↗

arXiv2024

Polynomial invariants for rooted trees related to their random destruction

We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site percolation on rooted trees. We exhibit recursion formulas for the invariants and identities relating $P$, $S$, and $M$. The main result states that the invariants $P$ and $S$ are complete, that is they distinguish rooted trees (in fact, even rooted forests) up to isomorphism. The proof method relies on the obtained recursion formulas and on irreducibility of the polynomials in suitable unique factorization domains. For $A$, we provide counterexamples showing that it is not complete, although that question remains open for the trivariate invariant $M$.

math.CO↗

arXiv2024

A bijection for the evolution of $B$-trees

A $B$-tree is a type of search tree where every node (except possibly for the root) contains between $m$ and $2m$ keys for some positive integer $m$, and all leaves have the same distance to the root. We study sequences of $B$-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with $B$-trees.

math.CO↗

arXiv2022

A Modification of the Random Cutting Model

We propose a modification to the random destruction of graphs: Given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.

math.PR↗

arXiv2021

A Semiring Structure for Generalised Pólya Urns

We define the notions of disjoint unions and products for generalised Pólya urns, proving that this turns the set of isomorphism classes of urns into a commutative semiring. The set of square matrices up to similarity by a permutation matrix is also a commutative semiring under the operations of direct sum and Kronecker sum, and we prove that assigning to an urn its intensity matrix leads to a morphism of semirings. Moreover, we show that a second semiring morphism exists, sending intensity matrices to their spectra. This, together with the existence of the first morphism has implications for the asymptotic behaviour of product urns, which are discussed.

math.PR↗