Related papers: Algebraic area enumeration for open lattice walks
Feller's book An Introduction to Probability Theory and Its Application discusses statistics corresponding to sequences of coin tosses, with a dollar being won or lost depending on the outcome of each toss. This is equivalent to analyzing…
We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.
We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding…
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the number of hyperbolic lines minus the number of…
Max-plus algebra is a kind of idempotent semiring over $\mathbb{R}_{\max}:=\mathbb{R}\cup\{-\infty\}$ with two operations $\oplus := \max$ and $\otimes := +$.In this paper, we introduce a new model of a walk on one dimensional lattice on…
The algebraic path problem provides a general setting for shortest path algorithms in optimization and computer science. This work extends the algebraic path problem to networks equipped with input and output boundaries. We show that the…
We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z^2, and always stay in the quadrant x >= 0, y >= 0. We first give a criterion which guarantees that the length generating…
For any positive integer $n$ along with parameters $\alpha$ and $\nu$, we define and investigate $\alpha$-shifted, $\nu$-offset, floor sequences of length $n$. We find exact and asymptotic formulas for the number of integers in such a…
We introduce the arithmetic width of a convex body, defined as the number of distinct values a linear functional attains on the lattice points within the body. Arithmetic width refines lattice width by detecting gaps in the lattice point…
The distribution of the first positive position reached by a random walker starting at the origin is central to the analysis of extremes and records in one-dimensional random walks. In this work, we present a detailed and self-contained…
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich…
We study the path behavior of the symmetric walk on some special comb-type subsets of ${\mathbb Z}^2$ which are obtained from ${\mathbb Z}^2$ by generalizing the comb having finitely many horizontal lines instead of one.
We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the…
This is a survey of results in the enumeration of lattice paths.
We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability $\alpha$ and $1-\alpha$, respectively) and starting from the origin. We show that, almost surely, the…
Gessel walks are lattice walks in the quarter plane $\set N^2$ which start at the origin $(0,0)\in\set N^2$ and consist only of steps chosen from the set $\{\leftarrow,\swarrow,\nearrow,\to\}$. We prove that if $g(n;i,j)$ denotes the number…
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial…
We use a discrete worldline representation in order to study the continuum limit of the one-loop expectation value of dimension two and four local operators in a background field. We illustrate this technique in the case of a scalar field…
We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the…
Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the…