Function Series, Catalan Numbers, and Random Walks on Trees
نویسندگان
چکیده
The delight of finding unexpected connections is one of the rewards of studying mathematics. In this talk, based on joint work with Ibtesam Bajunaid, Joel Cohen, and David Singman, I will discuss the connections that link the following seven superficially unrelated entities: (A) A function of the sort that calculus textbooks often use to show that a continuous function need not have a derivative at each point:
منابع مشابه
The distributions of some characteristics of random walks and related combinatorial identities
In this talk, we show an interesting fact that a quarter of paths of random walks of any length n(≥ 2) have two maximums. Moreover, it holds that the asymptotic distribution of number of maximums of paths obeys the geometric distribution of parameter 1/2. Furthermore, we provide the generating function for counting number of paths jointly with number k of maximums and number l of minimums. This...
متن کاملWalks on the slit plane
In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line H = {(k, 0), k ≤ 0}. We call them walks on the slit plane. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable generating function is algebraic of degree 8. Moreover, for any point (i, j), the length...
متن کاملA PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کاملRandom Walks on Infinite Graphs and Groups — a Survey on Selected Topics
Contents 1. Introduction 2 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 C. Random walks on groups 5 D. Group-invariant random walks on graphs 6 E. Harmonic and superharmonic functions 6 3. Spectral radius, amenability and law of large numbers 6 A. Spectral radius, isoperimetric inequalities and growth 6 B. Law of large numbers 9 ...
متن کاملUnsolved Problems Concerning Random Walks on Trees
We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on Galton-Watson trees monotonic in the drift parameter? These random walks have been used in Monte-Carlo algorithms for sampling from the vertices of a tree; in general, their behavior reflects the size and reg...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- The American Mathematical Monthly
دوره 112 شماره
صفحات -
تاریخ انتشار 2005