S-Motzkin paths with catastrophes and air pockets

So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.Comment: Very early version of the material; suggestions for extensions/improvements would be very welcom

Grand zigzag knight's paths

We study the enumeration of different classes of grand knight's paths in the plane. In particular, we focus on the subsets of zigzag knight's paths subject to constraints. These constraints include ending at ordinate 0, bounded by a horizontal line, confined within a tube, among other considerations. We present our results using generating functions or direct closed-form expressions. We derive asymptotic results, finding approximations for quantities such as the probability that a zigzag knight's path stays in some area of the plane, or for the average of the final height of such a path. Additionally, we exhibit some bijections between grand zigzag knight's paths and some pairs of compositions.Comment: 21 pages, 9 figure

Height of walks with resets, the Moran model, and the discrete Gumbel distribution

In this article, we consider several models of random walks in one or several dimensions, additionally allowing, at any unit of time, a reset (or "catastrophe") of the walk with probability $q$. We establish the distribution of the final altitude. We prove algebraicity of the generating functions of walks of bounded height $h$ (showing in passing the equivalence between Lagrange interpolation and the kernel method). To get these generating functions, our approach offers an algorithm of cost $O(1)$, instead of cost $O(h^3)$ if a Markov chain approach would be used. The simplest nontrivial model corresponds to famous dynamics in population genetics: the Moran model. We prove that the height of these Moran walks asymptotically follows a discrete Gumbel distribution. For $q=1/2$, this generalizes a model of carry propagation over binary numbers considered e.g. by von Neumann and Knuth. For generic $q$, using a Mellin transform approach, we show that the asymptotic height exhibits fluctuations for which we get an explicit description (and, in passing, new bounds for the digamma function). We end by showing how to solve multidimensional generalizations of these walks (where any subset of particles is attributed a different probability of dying) and we give an application to the soliton wave model

Stochastic Resetting and Applications

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate $r$, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate $r$. We then generalise to an arbitrary stochastic process (e.g. L\'evy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.Comment: 68 pages, Topical Review accepted version to appear in Journal of Physics A: Mathematical and Theoretical 202

Lattice paths with catastrophes

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not have the same intuitive probabilistic behaviour as classical Dyck paths (the typical properties of which are strongly related to Brownian motion theory), and this article quantifies some relations between these two types of paths. We give a bijection with some other lattice paths and a link with a continued fraction expansion. Furthermore, we prove several formulae for related combinatorial structures conjectured in the On-Line Encyclopedia of Integer Sequences. Thanks to the kernel method and via analytic combinatorics, we provide the enumeration and limit laws of these "lattice paths with catastrophes" for any finite set of jumps. We end with an algorithm to generate such lattice paths uniformly at random

Lattice paths with catastrophes

International audienc

Combinatorics of lattice paths and tree-like structures

Zusammenfassung in deutscher SpracheThis thesis is concerned with the enumerative and asymptotic analysis of directed lattice paths and tree-like structures. We introduce several new models and analyze some of their characterizing parameters, such as the number of returns to zero, or their average height and final altitude. The key tool in this context is the concept of generating functions. Their algebraic and analytic properties will help us to solve the enumeration problems. The methods and many other helpful theorems will be presented in the first part. Due to these methods this thesis belongs to the field of analytic combinatorics. The second part is dedicated to the study of directed lattice paths. Its first chapter treats the half-normal distribution, and presents a scheme for generating functions leading to such a distribution. We also state applications of this result in the theory of lattice paths. The next chapter continues the work of Cyril Banderier and Philippe Flajolet, and extends their work to the case when a boundary reflecting or absorbing condition is added to the classical models. The subsequent chapter then deals with a different family of paths: lattice paths below a line of rational slope. This work deals with the delicate problem of deriving asymptotic results for generating functions with a periodic support. It also answers an open problem by Donald E. Knuth on the asymptotics of such paths. The final chapter of this part deals with another model: lattice paths with catastrophes, which are jumps from any altitude to the x-axis. The third part treats the analysis of trees and tree-like structures. In the initial chapter we treat Pólya trees, which are unlabeled rooted trees. We present a new interpretation as Galton-Watson trees with many small forests. In the subsequent chapter we solve the counting problem of compacted trees of bounded right-height. Most trees contain redundant information in form of repeated occurrences of the same subtree. These trees can be compacted by representing each occurrence only once. The positions of the removed subtrees will be remembered by pointers which point to the common subtree. Such structures are known as directed acyclic graphs. The fourth and final part treats applications of analytic combinatorics to number theory. We study the exact divisibility of binomial coefficients by powers of primes by means of generating functions and singularity analysis.Die vorliegende Dissertation beschäftigt sich mit der analytischen und enumerativen Analyse von gerichteten Gitterwegen und baumartigen Strukturen. Es werden verschiedene, neue Modelle vorgestellt und einige ihrer charakterisierenden Parameter, wie unter anderem die Anzahl der Berührungen der x-Achse, oder ihre durchschnittliche und finale Höhe, untersucht. Das wichtigste Werkzeug in diesem Kontext sind erzeugende Funktionen. Die vorliegenden Ergebnisse beruhen zum Großteil auf ihren algebraischen und analytischen Eigenschaften, wie ihrer Singularitätsstruktur. Aus diesem Grund ist die vorliegende Arbeit dem Feld der analytischen Kombinatorik zuzuordnen. Eine Einführung in dieses Gebiet wird im ersten Teil dieser Arbeit gegeben. Der zweite Teil behandelt das Thema der gerichteten Gitterwege. Sein erstes Kapitel ist der Halbnormalverteilung gewidmet. Es wird eine neue Methode zur Charakterisierung von bivariaten erzeugenden Funktionen, in denen ein Parameter markiert wurde, der dieser Grenzverteilung gehorcht, präsentiert. Am Ende werden natürliche Vorkommen dieser Situation vorgestellt. Das folgende Kapitel löst ein offenes Problem von Donald E. Knuth über die Asymptotik von Wegen unter einer Geraden mit rationaler Steigung. Die Lösung benötigt die Behandlung von periodischen Trägern von erzeugenden Funktionen, welche zu periodischen Singularitätsstrukturen führen. Das anschließende Kapitel präsentiert aufbauend auf der Arbeit von Cyril Banderier und Philippe Flajolet ein neues Modell: das "reflection-absorption model". Dieses erlaubt die Modellierung einer reflektierenden oder absorbierenden Randbedingung. Das letzte Kapitel dieses Teils behandelt ein weiteres neues Modell für Gitterwege, in dem "Katastrophen" eingeführt werden. Dies sind Sprünge von beliebiger Höhe zur x-Achse. Der dritte Teil handelt von Bäumen und baumartigen Strukturen. Zunächst wird eine neue Interpretation von Pólya Bäumen (unmarkierten Wurzelbäumen) vorgestellt, welche diese als Galton-Watson Bäume charakterisiert, an die viele kleine Wälder angehängt werden. Im darauffolgenden Kapitel wird die Kompaktifizierung von binären Bäumen behandelt. Dies führt zu baumartigen Strukturen, den sogenannten "directed acyclic graphs". Ein kompaktifizierter Baum ist ein Baum in dem jeder Teilbaum eindeutig ist und mehrfach auftretende Teilbäume durch Zeiger ersetzt wurden. Durch die Modellierung solcher Objekte mittels exponentiell erzeugender Funktionen wird das asymptotische Abzählproblem für kompaktifizierte Bäume mit beschränkter rechtsseitiger Höhe gelöst. Im vierten und letzten Teil wird ein neuer Themenschwerpunkt behandelt: Die Anwendung der analytischen Kombinatorik in der Zahlentheorie. Hier wird die exakte Teilbarkeit von Binomialkoeffizienten durch Potenzen von Primzahlen untersucht.34