Related papers: Walks with Small Steps in the 4D-Orthant
This paper is the first application of the compensation approach to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of…
In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the non positive horizontal half-axis. We call them "walks on the slit plane". We count them by their length,…
Photonic implementations of unitary processes on lattice structures, such as quantum walks, have been demonstrated across various architectures. However, few platforms offer the combined advantages of scalability, reconfigurability, and the…
Scaling mobility patterns have been widely observed for animals. In this paper, we propose a deterministic walk model to understand the scaling mobility patterns, where walkers take the least-action walks on a lattice landscape and prey.…
A family of walks confined to the first orthant whose defining stepset has drift outside of the region can be challenging to sample uniformly at random for large lengths. We address this by generalizing the 2D walk sampler of Lumbroso et…
We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive…
A random walk generated by a sum of independent identity distributed random variables with positive expectation is considered. The limiting distributions for the first- passage -time of a step-function boundary are derived.
Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if…
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…
We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$. Previous work in this area has established asymptotics for the…
The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their…
The aim of this article is to introduce a unified method to obtain explicit integral representations of the trivariate generating function counting the walks with small steps which are confined to a quarter plane. For many models, this…
We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogenous random walks in the plane which interlace two…
We investigate the cutoff effects in 2-d lattice O(N) models for a variety of lattice actions, and we identify a class of very simple actions for which the lattice artifacts are extremely small. One action agrees with the standard action,…
We realize a pair of simultaneous ten-step one-dimensional quantum walks with two walkers sharing coins, which we prove is analogous to the ten-step two-dimensional quantum walk with a single walker holding a four-dimensional coin. Our…
We make and generalize the observation that summing of probability amplitudes of a discrete-time quantum walk over partitions of the walking graph consistent with the step operator results in a unitary evolution on the reduced graph which…
We give a summary of recent progress on the signed area enumeration of closed walks on planar lattices. Several connections are made with quantum mechanics and statistical mechanics. Explicit combinatorial formulae are proposed which rely…
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…
The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time…