Related papers: Existence and bounds of growth constants for restr…
We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L, L)$, and are entirely contained in the square $[0, L] \times [0, L]$ on the square lattice ${\mathbb Z}^2$. The number of distinct walks…
We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called 'height…
Oriented self-avoiding walks (OSAWs) on a square lattice are studied, with binding energies between steps that are oriented parallel across a face of the lattice. By means of exact enumeration and Monte Carlo simulation, we reconstruct the…
Building on a work by Alm, we consider a model of weighted self-avoiding walks on a lattice and develop a method for computing upper bounds on the corresponding weighted connective constant, which we implement in a publicly available…
We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $\lambda_S^{L^2}.$ We…
We study self-avoiding walks on restricted square lattices, more precisely on the lattice strips $\mathbb{Z} \times \{-1,0,1\}$ and $\mathbb{Z}\times \{-1,0,1,2\}$. We obtain the value of the connective constant for the $\mathbb{Z} \times…
We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact…
We find the generating function of self-avoiding walks and trails on a semi-regular lattice called the $3.12^2$ lattice in terms of the generating functions of simple graphs, such as self-avoiding walks, polygons and tadpole graphs on the…
The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors…
We calculate improved lower bounds for the connective constants for self-avoiding walks on the square, hexagonal, triangular, $(4.8^2)$, and $(3.12^2)$ lattices. The bound is found by Kesten's method of irreducible bridges. This involves…
Despite its elementary definition, the self-avoiding walk (SAW) poses notoriously hard enumerative problems: exact connective constants are known for only a handful of infinite graphs, notably the honeycomb lattice \cite{ds}. We establish a…
We define a new family of self-avoiding walks (SAW) on the square lattice, called weakly directed walks. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating…
Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was…
We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the…
Various types of walks on complex networks have been used in recent years to model search and navigation in several kinds of systems, with particular emphasis on random walks. This gives valuable information on network properties, but…
Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are…
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the…
Self-avoiding walks (SAWs) were introduced in chemistry to model the real-life behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point. In mathematics, a…
Two closely related models of oriented self-avoiding walks (OSAWs) on a square lattice are studied. We use the pruned-enriched Rosenbluth method to determine numerically the phase diagram. Both models have three phases: a tight-spiral phase…
The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the…