Number Of Self-avoiding Walks Research Articles

A general bridge theorem for self-avoiding walks

Let X be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on X is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number of bridges of length n starting at a vertex o of X grow exponentially in n and the bases of these growth rates are called connective constant and bridge constant, respectively. We show that for any graph height function h the connective constant of the graph is equal to the maximum of the two bridge constants given by increasing and decreasing bridges with respect to h. As a concrete example, we apply this result to calculate the connective constant of the Grandparent graph.

Self-avoiding walks and connective constants in clustered scale-free networks

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 self-avoiding walks (SAWs) may be more suitable than unrestricted random walks to study long-distance characteristics of complex systems. Here we study SAWs in clustered scale-free networks, characterized by a degree distribution of the form P(k)∼k^{-γ} for large k. Clustering is introduced in these networks by inserting three-node loops (triangles). The long-distance behavior of SAWs gives us information on asymptotic characteristics of such networks. The number of self-avoiding walks, a_{n}, has been obtained by direct enumeration, allowing us to determine the connective constant μ of these networks as the large-n limit of the ratio a_{n}/a_{n-1}. An analytical approach is presented to account for the results derived from walk enumeration, and both methods give results agreeing with each other. In general, the average number of SAWs a_{n} is larger for clustered networks than for unclustered ones with the same degree distribution. The asymptotic limit of the connective constant for large system size N depends on the exponent γ of the degree distribution: For γ>3,μ converges to a finite value as N→∞; for γ=3, the size-dependent μ_{N} diverges as lnN, and for γ<3 we have μ_{N}∼N^{(3-γ)/2}.

The set of connective constants of Cayley graphs contains a Cantor space

The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.

Structural characterization of ice polymorphs from self-avoiding walks

Topological properties of crystalline ice structures are studied by means of self-avoiding walks on their H-bond networks. The number of self-avoiding walks, Cn, for eight ice polymorphs has been obtained by direct enumeration up to walk length n=27. This has allowed us to determine the ‘connective constant’ or effective coordination number μ of these structures as the limit of the ratio Cn/Cn-1 for large n. This structure-dependent parameter μ is related with other topological characteristics of ice polymorphs, such as the mean and minimum ring size, or the topological density of network sites. A correlation between the connective constant and the configurational entropy of hydrogen-disordered ice structures is discussed.

Position and orientation distributions for locally self-avoiding walks in the presence of obstacles

This paper presents a new approach to study the statistics of lattice random walks in the presence of obstacles and local self-avoidance constraints (excluded volume). By excluding sequentially local interactions within a window that slides along the chain, we obtain an upper bound on the number of self-avoiding walks (SAWs) that terminate at each possible position and orientation. Furthermore we develop a technique to include the effects of obstacles. Thus our model is a more realistic approximation of a polymer chain than that of a simple lattice random walk, and it is more computationally tractable than enumeration of obstacle-avoiding SAWs. Our approach is based on the method of the lattice motion-group convolution. We develop these techniques theoretically and present numerical results for 2-D and 3-D lattices (square, hexagonal, cubic and tetrahedral/diamond). We present numerical results that show how the connectivity constant μ changes with the length of each self-avoiding window and the total length of the chain. Quantities such as 〈R〉 and others such as the probability of ring closure are calculated and compared with results obtained in the literature for the simple random walk case.

On the Number of Self-Avoiding Walks on Hyperbolic Lattices

On the Number of Self-Avoiding Walks on Hyperbolic Lattices

How Many Needles are in a Haystack, or How to Solve #P-Complete Counting Problems Fast

We present two randomized entropy-based algorithms for approximating quite general #P-complete counting problems, like the number of Hamiltonian cycles in a graph, the permanent, the number of self-avoiding walks and the satisfiability problem. In our algorithms we first cast the underlying counting problem into an associate rare-event probability estimation, and then apply dynamic importance sampling (IS) to estimate efficiently the desired counting quantity. We construct the IS distribution by using two different approaches: one based on the cross-entropy (CE) method and the other one on the stochastic version of the well known minimum entropy (MinxEnt) method. We also establish convergence of our algorithms and confidence intervals for some special settings and present supportive numerical results, which strongly suggest that both ones (CE and MinxEnt) have polynomial running time in the size of the problem.

Self-avoiding walks crossing a square

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 is known to grow as $\lambda^{L^2+o(L^2)}$. We estimate $\lambda = 1.744550 \pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < \lambda < 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$. We also consider the model in which a weight or {\em fugacity} $x$ is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For $x < 1/\mu$ the average length of a SAW grows as $L$, while for $x > 1/\mu$ it grows as $L^2$. Here $\mu$ is the growth constant of unconstrained SAW in ${\mathbb Z}^2$. For $x = 1/\mu$ we provide numerical evidence, but no proof, that the average walk length grows as $L^{4/3}$. We also consider Hamiltonian walks under the same restriction. They are known to grow as $\tau^{L^2+o(L^2)}$ on the same $L \times L$ lattice. We give precise estimates for $\tau$ as well as upper and lower bounds, and prove that $\tau < \lambda.$

On the Ergodicity of Dynamic Monte Carlo Simulations of Multichain or Star Systems

AbstractSummary:The behavior of complex polymer structures, e.g., star and comb polymers or shells of polymer micelles, is often studied by dynamic Monte Carlo simulations. The algorithm, which is based on a sequence of independent steps, each of them consisting in dissolving and regrowing a randomly chosen tethered chain by the configuration‐bias Monte Carlo (CBMC) method, is considered. During each step, the remaining self‐avoiding walks (SAWs), which occupy the space, create geometrical restriction for the new SAW and hinder certain conformations. Hence, the reconstruction of the SAW under consideration depends on conformations of the other SAWs forming the system, and therefore, it is not directly evident whether the a priori ergodicity of SAW for a single untethered chain has been retained in the final algorithm for the whole multichain system. The proof of ergodicity of this type of simulations for an arbitrary number of SAWs tethered to the convex core is presented.2D scheme of the alignment of the SAWs.image2D scheme of the alignment of the SAWs.

The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

Self-avoiding walk on a three-dimensional Manhattan lattice

We have extended the definition of the Manhattan lattice from two-dimensional to three-dimensional (3D) spaces. The number of self-avoiding walks on the 3D Manhattan lattice, Cn, and their mean-square end-to-end distances, Rn2, were counted exactly up to 31 and 30 steps, respectively. Analysis using the method of the Dlog?Pad? approximant gave the exponents ? = 1.1615?0.0002 and ? = 0.5870?0.0025, which are in good agreement with corresponding values for self-avoiding walks on the ordinary 3D lattice. This result suggests that self-avoiding walks on the 3D Manhattan lattice belong to the same universality class as self-avoiding walks on the ordinary 3D lattice.

Self-avoiding walks and connective constants in zeolitic frameworks

The number of self-avoiding walks, un, in several zeolitic frameworks has been calculated up to distances n higher than 20 steps. This allows us to estimate the 'connective constant' or effective coordination number of these structures as the limit of the ratio un/un-1 for large n. This parameter is correlated with the density of tetrahedra in the zeolite networks, and is shown to be very useful for characterizing the network connectivity, as well as quantifying the influence of the framework topology on the Si, Al atom ordering in these materials.

Asymptotic behaviour of the number of self-avoiding walks on finitely ramified fractals

We study self-avoiding walks (SAWs) on the checkerboard (CB) family of fractals, each member of which can be labelled by an odd integer b (3<or=b< infinity ), so that the fractal dimension df tends to the Euclidean value 2 when b to infinity . By applying the exact renormalization-group method (for b=3,5 and 7), and the Monte Carlo renormalization-group method (for a sequence of fractals with 5<or=b<or=81), we have calculated the critical exponent gamma , associated with the total number of distinct SAWs. It turns out that gamma , being always larger than the corresponding Euclidean value 43/32 increases monotonically with b. In order to learn the asymptotic behaviour of gamma for large b, we have applied the finite-size scaling (FSS) method (based on previous exact results for wedges of the two-dimensional Euclidean lattices), and thereby we have shown that Applying the same method (FSS), we have also demonstrated that the critical exponent nu , associated with the mean-squared end-to-end distance of SAWs, tends to the Euclidean value 3/4 from below, when b to infinity . The obtained results extend, and in a way weld together, the previous studies of SAWs various families of finitely ramified fractals.

Exact enumeration of self-avoiding walks on lattices with random site energies

The self-avoiding random walk on lattices with quenched random site energies is studied using exact enumeration in d=2 and 3. For each configuration we compute the size R and energy E of the minimum-energy self-avoiding walk (SAW). Configuration averages yield the exponents \ensuremath{\nu} and \ensuremath{\chi}, defined by ${\mathit{R}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}${\mathit{N}}^{2\ensuremath{\nu}}$ and \ensuremath{\delta}${\mathit{E}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}${\mathit{N}}^{2\mathrm{\ensuremath{\chi}}}$. These calculations indicate that \ensuremath{\nu} is significantly larger than its value in the pure system. Finite-temperature studies support the notion that the system is controlled by a zero-temperature fixed point. Consequently, exponents obtained from minimum-energy SAW's characterize the properties of finite temperature SAW's on disordered lattices.

Self-avoiding walk connectivity constant and theta point on percolating lattices

The average connectivity constant mu of self-avoiding walks (SAWs) is obtained from exact enumeration of SAWs on Monte Carlo generated percolating clusters in a randomly diluted square lattice. For averages over the (infinite) percolating cluster, mu decreases almost linearly with bond dilution (1-p), where p is the bond occupation concentration. The authors find mu (pc)=1.31+or-0.03 at the percolation threshold pc and could not detect any significant difference between mu (pc) and pc mu (1). The variation of theta -point for SAWs on the same lattice with dilution is also estimated, analysing the partition function zeros. Within the limited accuracy of their analysis, its variation with dilution is observed as being quite weak and the theta -point increases somewhat (compared to pure lattice value) near pc; they find a non-vanishing theta point (Ktheta (pc) equivalent to 0.59, where K0=J/k theta ) on the square lattice percolation cluster at pc.

Exact series studies of self-avoiding walks on two-dimensional critical percolation clusters

The author presents results of exact series studies of self-avoiding walks on percolation clusters on the square lattice performed very close to the percolation threshold and estimates the critical exponents nu and gamma defined by the disorder averages of the mean square radius of gyration or the end-to-end distance and the number of self-avoiding walks, respectively. The self-avoiding walk configurations are enumerated exactly up to 35 steps. The result for nu indicates a large increase compared with the self-avoiding walks on fully occupied lattice. The result for gamma , however, indicates a behaviour similar to that on a fully occupied lattice. The result for nu is completely different from that of Lee, Nakanishi and Kim (1989).

Monotonicity of the number of self-avoiding walks

It is shown that the numberc n of self-avoiding walks of lengthn in ℤ d is an increasing function ofn.

Monte Carlo study of self-avoiding walks on a critical percolation cluster.

We present the results of Monte Carlo simulation for self-avoiding walks on a percolation cluster for square and simple cubic lattices performed very close to the percolation thresholds and estimate critical exponents \ensuremath{\nu} and \ensuremath{\gamma} defined by the disorder averages of the mean-square radius of gyration and the number of self-avoiding walks, respectively. Our results for \ensuremath{\nu} indicate a behavior rather similar to the self-avoiding walks on fully occupied lattice unlike the large increase in \ensuremath{\nu} reported in the only previous work of this kind by Kremer given for the diamond lattice. Our results for \ensuremath{\gamma} suggest an increase from the full lattice value just at ${p}_{c}$ in three dimensions, while in two dimensions the asymptotic behavior appears to be similar to that of the ordinary self-avoiding walks. We also consider the possible crossover from fractal to Euclidean behavior and discuss the reasons why no crossover scaling is observed for the mean-square radius of gyration in the present calculation.

Examination of the theta -point from exact enumeration of self-avoiding walks. II

Examination of the theta -point from exact enumeration of self-avoiding walks. II

Study of the θ point by enumeration of self-avoiding walks on the triangular lattice

The author reports series expansion analyses of self-avoiding walks with nearest neighbour bond interactions. The estimates 2 nu t=1.07+or-0.05 and phi =0.64+or-0.05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end-to-end distance data up to 16 steps on the triangular lattice.