Related papers: Walks in the quadrant with interacting boundaries …
Relations between the mean values of distributions of flipped spins on periodic Heisenberg XX chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered…
This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel.…
This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. The main difference to classical walks is that its nondeterministic steps consist of sets of steps from a predefined set such that all possible…
It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results…
A new characterization of the Lovasz theta function is provided by relating it to the (weighted) walk-generating function, thus establishing a relationship between two seemingly quite distinct concepts in algebraic graph theory. An…
A method is described to count simple diagonal walks on $\mathbb{Z}^2$ with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller…
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…
This article introduces a model for interacting vertex-reinforced random walks, each taking values on a complete sub-graph of a locally finite undirected graph. The transition probability for a walk to a given vertex depends on the…
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$,…
Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schr\"{o}der, and Motzkin number sequences…
We consider singular (aka genus $0$) walks in the quarter plane and their associated generating functions $Q(x,y,t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables $x$ and $y$) and…
A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their…
In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but…
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is…
We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only…
Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at…
The process by which one may take a discrete model of a biophysical process and construct a continuous model based on it is of mathematical interest as well as being of practical use. In this paper, we first study the singular limit of a…
The article provides an explicit algebraic expression for the generating function of walks on graphs. Its proof is based on the scattering theory for the differential Laplace operator on non-compact graphs.
Full counting statistics concerns the stochastic transport of electrons in mesoscopic structures. Recently it has been shown that the charge transport statistics for non-interacting electrons in a two-terminal system is always generalized…
The question of classifying the nature of the generating functions of restricted lattice walks has enjoyed much attention in past years. We prove that a certain class of octant walks have a D-finite generating function using the theory of…