arXiv subjects

David Fabian

Publications and source records attributed to David Fabian.

8 recordsLinked to original sources

arXiv2026

A logarithmic phase singularity at the heart of Landau-Zener transitions

Three ingredients of the elementary Landau-Zener problem determine the familiar expression $a_{LZ}\equiv\exp\left[-\pi/(2\epsilon)\right]$ for the asymptotic value of the probability amplitude for remaining in the initial level: (i) A wave whose phase is determined by the product of a contour integral over a simple pole at the origin of the complex plane and the inverse of twice the scaled chirp parameter $\epsilon$. (ii) An asymptotic limit of the associated path connecting the points $\pm 1$ along the real axis and circumventing the pole in the upper half-plane, and (iii) a half-circle in the lower half plane enclosing together with the asymptotic path the pole. The Cauchy theorem immediately provides us with the value $\ii\pi$ of the asymptotic contour, and thus with $a_{LZ}$. Our analysis demonstrates not only that $a_{LZ}$ is the consequence of a logarithmic phase singularity but also explains why the Markov approximation also leads to $a_{LZ}$.

quant-ph↗

arXiv2026

Graph bootstrap percolation -- a discovery of slowness

Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of $K_n$. Given an initial set $G\subseteq K_n$ of infected edges, the transmission of the virus is governed by a fixed graph $H$: in each round of the process any edge $e$ of $K_n$ that is the last uninfected edge in a copy of $H$ in $K_n$ gets infected as well. Once infected, edges remain infected forever. The process was introduced by Bollob\'as in 1968 in the context of weak saturation and has since inspired a vast array of beautiful mathematics. The main focus of this survey is the extremal question of how long the infection process can last before stabilising. We give an exposition of our recent systematic study of this maximum running time and the influence of the infection rule $H$. The topic turns out to possess a wide variety of interesting behaviour, with connections to additive, extremal and probabilistic combinatorics. Along the way we encounter a number of surprises and attractive open problems.

math.CO↗

arXiv2026

Advanced Quantum Communication and Quantum Networks -- From basic research to future applications

Classical communication is the basis for many of our current and future technologies, such as mobile phones, video conferences, autonomous vehicles and particularly the internet. In contrast, quantum communication is governed by the laws of quantum mechanics. Due to this fundamental difference, it might offer enormous benefits for security applications, more precise measurements, faster computations, and many other fields of application by interconnecting different quantum devices, such as quantum sensors, quantum computers, or quantum memories. This review provides an overview of the specific properties of quantum information networks. This includes the interfaces between the classical and the quantum regime, the transmission of the quantum information by physical implementations, and potential future applications of quantum networks. We aim to provide a starting point based on fundamental concepts of quantum information processing for further research on a future quantum internet.

quant-ph↗

arXiv2025

Slow graph bootstrap percolation III: Chain constructions

For graphs $H$, we study the extremal function $M_H(n)$ which is the maximum running time (until stabilisation) of an $H$-bootstrap percolation process on $n$ vertices. Building on previous work in the clique case $H=K_k$, we develop a general framework of chain constructions. We demonstrate the flexibility of this framework by applying several variations of the method to give lower bounds on $M_H(n)$ for a wide variety of different graphs $H$ including dense graphs, random graphs and complete bipartite graphs. In particular, we focus on the question of whether $M_H(n)$ is (almost) quadratic or not and our lower bounds develop connections with additive combinatorics, utilising constructions of sets free of solutions to certain linear equations. Finally, our lower bounds are complemented by upper bounds which connect $M_H(n)$ to other problems in extremal graph theory such as the Ruzsa-Szemer\'edi (6,3)-Theorem.

math.CO↗

arXiv2023

Slow graph bootstrap percolation II: Accelerating properties

For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap process on $G$ is the process which starts with $G$ and, at every time step, adds any missing edges on the vertices of $G$ that complete a copy of $H$. This process eventually stabilises and we are interested in the extremal question raised by Bollob\'as of determining the maximum running time (number of time steps before stabilising) of this process over all possible choices of $n$-vertex graph $G$. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted $M_H(n)$, and its dependence on properties of the graph $H$. Our focus is on $H$ which define relatively fast bootstrap processes, that is, with $M_H(n)$ being at most linear in $n$. We study the graph class of trees, showing that one can bound $M_T(n)$ by a quadratic function in $v(T)$ for all trees $T$ and all $n$. We then go on to explore the relationship between the running time of the $H$-process and the minimum vertex degree and connectivity of $H$.

math.CO↗

arXiv2023

Slow graph bootstrap percolation I: Cycles

Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$\emph{-bootstrap percolation process} on $G$ is defined to be the sequence of graphs $G_i$, $i\geq 0$ which starts with $G_0 := G$ and in which $G_{i+1}$ is obtained from $G_i$ by adding every edge that completes a copy of $H$. We are interested in $M_H(n)$ which is the maximum number of steps, over all $n$-vertex graphs $G$, that this process takes to stabilise. We determine this maximum running time precisely when $H$ is a cycle, giving the first infinite family of graphs $H$ for which an exact solution is known. We find that $M_{C_k}(n)$ is of order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly though, the function exhibits different behaviour depending on the parity of $k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is determined by the Frobenius number of a certain numerical semigroup depending on $k$.

math.CO↗

arXiv2022

Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.

math.CO↗

arXiv2019

On strong infinite Sidon and $B_h$ sets and random sets of integers

A set of integers $S \subset \mathbb{N}$ is an $α$-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on $α$, more specifically if $| (x+w) - (y+z) | \geq \max \{ x^α,y^α,z^α,w^α\}$ for every $x,y,z,w \in S$ satisfying $\max \{x,w\} \neq \max \{y,z\}$. We obtain a new lower bound for the growth of $α$-strong infinite Sidon sets when $0 \leq α< 1$. We also further extend that notion in a natural way by obtaining the first non-trivial bound for $α$-strong infinite $B_h$ sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or $B_h$ set contained in a random infinite subset of $\mathbb{N}$. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rödl.

math.CO↗