arXiv subjects
Publications and source records attributed to Stephan Wagner.
We consider point spectrum traces in the Hofstadter model. We show how to recover the full quantum Hofstadter trace by integrating these point spectrum traces with the appropriate free density of states on the lattice. This construction is then generalized to the almost Mathieu operator and its n-th moments which can be expressed in terms of generalized Kreft coefficients.
A tanglegram consists of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. Tanglegrams are drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines, and the perfect matching inside the strip. If this can be done without any edges crossing, a tanglegram is called planar. We show that every non-planar tanglegram contains one of two non-planar 4-leaf tanglegrams as induced subtanglegram, which parallels Kuratowski's Theorem.
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a compactification of the space of countable (connected) rooted graphs, and use it to generalise the notion of Benjamini-Schramm convergence in order to allow for vertices of infinite degree in the limit object.
A Lucas sequence is a sequence of the general form $v_n = (ϕ^n - \barϕ^n)/(ϕ-\barϕ)$, where $ϕ$ and $\barϕ$ are real algebraic integers such that $ϕ+\barϕ$ and $ϕ\barϕ$ are both rational. Famous examples include the Fibonacci numbers, the Pell numbers, and the Mersenne numbers. We study the monoid that is generated by such a sequence; as it turns out, it is almost freely generated. We provide an asymptotic formula for the number of positive integers $\leq x$ in this monoid, and also prove Erdős-Kac type theorems for the distribution of the number of factors, with and without multiplicity. While the limiting distribution is Gaussian if only distinct factors are counted, this is no longer the case when multiplicities are taken into account.
In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalise strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.
We consider the generating function of the algebraic area of lattice walks, evaluated at a root of unity, and its relation to the Hofstadter model. In particular, we obtain an expression for the generating function of the n-th moments of the Hofstadter Hamiltonian in terms of a complete elliptic integral, evaluated at a rational function. This in turn gives us both exact and asymptotic formulas for these moments.
In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.
A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where $B_1,\ldots,B_k$ are the branches of the tree $T$ and $f(T)$ is a toll function. We prove a general central limit theorem for additive functionals of $d$-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of $d$-ary increasing trees, but many other examples (old and new) are covered as well.
A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random plane binary trees. This fact is used to derive a number of results on the shape of random tanglegrams, including theorems on the number of cherries and generally occurrences of subtrees, the root branches, the number of automorphisms, and the height. For each of these, we obtain limiting probabilities or distributions. Finally, we investigate the number of matched cherries, for which the limiting distribution is identified as well.
The aim of this paper is to provide an affirmative answer to a recent question by Bubeck and Linial on the local profile of trees. For a tree $T$, let $p^{(k)}_1(T)$ be the proportion of paths among all $k$-vertex subtrees (induced connected subgraphs) of $T$, and let $p^{(k)}_2(T)$ be the proportion of stars. Our main theorem states: if $p^{(k)}_1(T_n) \to 0$ for a sequence of trees $T_1,T_2,\ldots$ whose size tends to infinity, then $p^{(k)}_2(T_n) \to 1$. Both are also shown to be equivalent to the statement that the number of $k$-vertex subtrees grows superlinearly and the statement that the $(k-1)$th degree moment grows superlinearly.
In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $γ(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that induce a tree isomorphic to $B$. The inducibility of $B$ is $\limsup_{|T| \to \infty} γ(B,T)$. We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility $1$. We also formulate some open problems and conjectures on the inducibility. Finally, we present an application to crossing numbers of random tanglegrams.
We show that a tree of order $n$ has at most $O(5^{n/4})$ nonisomorphic subtrees, and that this bound is best possible. We also prove an analogous result for the number of nonisomorphic rooted subtrees of a rooted tree.
We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are "easier to reach" by a random walk, but "more difficult to get out of". We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behaviour or random walk on a graph to its eigenvalues in a new perspective.
Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.
We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erdős-Surányi sequence if every integer can be written in this form for some $n\in\mathbb{N}$ and choices of signs in infinitely many ways. We study the number of representations of a fixed integer, which can be written as a trigonometric integral, and obtain an asymptotic formula under a rather general scheme due to Roth and Szekeres. Our approach, which is based on Laplace's method for approximating integrals, can also be easily extended to find higher-order expansions. As a corollary, we settle a conjecture of Andrica and Ionaşcu on the number of solutions to the signum equation $\pm 1^k \pm 2^k \pm \dotsb \pm n^k = 0$.
For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer $n$. Moreover, we prove central limit theorems for the sum of digits, the Hamming weight (number of non-zero digits, which is a measure of efficiency) and the occurrences of a fixed digits in a random representation.
Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.