Related papers: Self-avoiding walks on cubic graphs and local tran…
A prototypical problem on which techniques for exact enumeration are tested and compared is the enumeration of self-avoiding walks. Here, we show an advance in the methodology of enumeration, making the process thousands or millions of…
Weakly self-avoiding walk (WSAW) is a model of simple random walk paths that penalizes self-intersections. On $\mathbb{Z}$, Greven and den Hollander proved in 1993 that the discrete-time weakly self-avoiding walk has an asymptotically…
Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum.…
In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space…
We show numerically that critical exponents for two-point interchain correlation of an infinite chain characterize those of finite chains in Self-Avoiding Walk (SAW) and Self-Avoiding Polygon (SAP) under a topological constraint. We…
A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain…
We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W_n(S), and rooted…
We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite…
We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $\beta \rightarrow \beta_c-$ of the probability…
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…
We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents $\nu$ and $2\Delta_4…
We prove several rigorous results about the asymptotic behaviour of the numbers of polygons and self-avoiding walks confined to a square on the square lattice. Specifically we prove that the dominant asymptotic behaviour of polygons…
A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be formed using the letters of the alphabet $V$ such that for every pair of vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. A…
An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced…
A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$. In this paper, we prove a fundamental result on equitable…
The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $ d $-dimensional lattice obeys hyperscaling. The combination $ p_N \left\langle R^2 \right\rangle^{ d/2}_N\mu^{ -N}, $ (where $ p_N $ is the number of…
We recently published [J. Phys A: Math. Theor. {\bf 45} 115202 (2012)] a new and more efficient implementation of a transfer-matrix algorithm for exact enumerations of self-avoiding polygons. Here we extend this work to the enumeration of…
We consider a model of semi-flexible interacting self-avoiding trails (sISAT's) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting…
We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically…
Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In…