Related papers: Some reflections on directed lattice paths
We establish a reflection principle for three lattice walkers and use this principle to reduce the enumeration of the configurations of three vicious walkers to that of configurations of two vicious walkers. In the combinatorial treatment…
We extend the active walker model to address the formation of paths on gradients, which have been observed to have a zigzag form. Our extension includes a new rule which prohibits direct descent or ascent on steep inclines, simulating…
We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are…
In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$.…
The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by…
Linear polymers adsorbing on a wall can be modelled by half-space self-avoiding walks adsorbing on a line in the square lattice, or on a surface in the cubic lattice. In this paper a numerical approach based on the GAS algorithm is used to…
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…
We investigate the surface adsorption transition of interacting self-avoiding square lattice trails onto a straight boundary line. The character of this adsorption transition depends on the strength of the bulk interaction, which induces a…
We investigate the dynamics of a one-dimensional asymmetric exclusion process with Langmuir kinetics and a fluctuating wall. At the left boundary, particles are injected onto the lattice; from there, the particles hop to the right. Along…
We consider planar lattice walks that start from (0,0), remain inthe first quadrant i, j >= 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic…
In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing…
We present refined enumeration formulas for lattice paths in $\mathbb{Z}^2$ with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the…
The paper is devoted to the study of lattice paths that consist of vertical steps $(0,-1)$ and non-vertical steps $(1,k)$ for some $k\in \mathbb Z$. Two special families of primary and free lattice paths with vertical steps are considered.…
We study a recently introduced ladder model which undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behaviour of the model is the same as that of the branching annihilating…
We study the pedestrian escape from an obscure corridor using a lattice gas model with two species of particles. One species, called passive, performs a symmetric random walk on the lattice, whereas the second species, called active, is…
We explore rectification phenomena in a system where two-dimensional random walkers interact with a funnel-shaped ratchet under two distinct classes of reflection rules. The two classes include the angle of reflection exceeding the angle of…
We study lattice walks in a Weyl chamber of type A with fixed or free end points. For lattice walk models with zero drift that may be counted by means of a reflection argument, we determine asymptotics for the number of such walks as their…
A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schr\"oder path. In this paper, we generalize…
Sums of walks for charged particles (e.g. Hofstadter electrons) on a square lattice in the presence of a magnetic field are evaluated. Returning loops are systematically added to directed paths to obtain the unrestricted propagators.…
We consider the reversible adsorption of dimers on a regular lattice, where adsorption occurs on a finite fraction of sites selected randomly. By comparing this system to the pure system where all sites are available for adsorption, we show…