Related papers: Partially Directed Snake Polyominoes
In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study…
L\'evy walks are random walk processes whose step-lengths follow a long-tailed power-law distribution. Due to their abundance as movement patterns of biological organisms, significant theoretical efforts have been devoted to identifying the…
We present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential. We consider three geometric constraints on the ends of the polymer and…
This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give…
We consider staircase polygons, counted by perimeter and sums of k-th powers of their diagonal lengths, k being a positive integer. We derive limit distributions for these parameters in the limit of large perimeter and compare the results…
The correlations in the motion of reptating polymers in their melt are investigated by means of kinetic Monte Carlo simulations of the three dimensional slithering snake version of the bond-fluctuation model. Surprisingly, the slithering…
This is the second report of our work on the construction of directed strongly regular graphs. In our previous work, we constructed a couple of infinite families of new directed strongly regular graphs on the sets of antiflags of partial…
Snake robots have showcased remarkable compliance and adaptability in their interaction with environments, mirroring the traits of their natural counterparts. While their hyper-redundant and high-dimensional characteristics add to this…
In this paper we consider a restricted class of convex polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any two pairs of cells can be connected by a monotone path making at most two turns (like…
For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a…
Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational…
We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence…
This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of…
We study the geometry associated with the kinematics of a planar robot known as the "three-segment snake," whose velocity distribution belongs to a class of (2,3,5) distributions. We discover that, under certain assumptions on its…
We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O$(n^5),$ for a series of length $n.$ We have extended the definition to three dimensions and produced series…
A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function $\G$ of directed animals on the square…
Limbless organisms such as snakes can navigate nearly all terrain. In particular, desert-dwelling sidewinder rattlesnakes (Crotalus cerastes) operate effectively on inclined granular media (such as sand dunes) that induce failure in…
Snakes can move through almost any terrain. Similarly, snake robots hold the promise as a versatile platform to traverse complex environments like earthquake rubble. Unlike snake locomotion on flat surfaces which is inherently stable, when…
Neural correlations during a cognitive task are central to study brain information processing and computation. However, they have been poorly analyzed due to the difficulty of recording simultaneous single neurons during task performance.…
Spider silks are remarkable materials characterized by superb mechanical properties such as strength, extensibility and lightweightedness. Yet, to date, limited models are available to fully explore sequence-property relationships for…