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Self-avoiding walk enumeration via the lace expansion
Journal of Physics A: Mathematical and Theoretical
Abstract
We introduce a new method for the enumeration of self-avoiding walks based on the lace expansion. We also introduce an algorithmic improvement, called the two-step method, for self-avoiding walk enumeration problems. We obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3: we enumerate 32-step self-avoiding polygons in d = 3, 26-step self-avoiding polygons in d = 4, 30-step self-avoiding walks in d = 3, and 24-step self-avoiding walks and polygons in all dimensions d 4. We analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8. We also provide major extensions of 1/d expansions for the connective constant and for two critical amplitudes.
... In 1992, MacDonald et al. [10] reached N max = 23, and in 2000 MacDonald et al. [11] reached N max = 26. In 2007, a combination of the lace expansion and the two-step method allowed for the enumeration of SAWs up to N max = 30 steps [12]. Recently, the length-doubling method [13] was presented which allowed enumerations to be extended up to N max = 36. ...
... We reduce the influence of this additional sub-leading correction by separately treating the sequences for even and odd N . See [12] for more detailed discussion on this point for the asymptotic behaviour of Z N on the SC lattice, which is also bipartite. ...
... Differential approximants are extremely effective at extracting information about critical exponents from the long series that have been obtained for two-dimensional lattice models, such as self-avoiding polygons [22] or walks on the square lattice [23]. However, differential approximants have been far less successful for the shorter series available for three-dimensional models such as SAWs on the simple cubic lattice [12,13]. For short series, it seems that corrections-to-scaling due to confluent corrections are too strong at the orders that can be reached to be able to reliably determine critical exponents. ...
Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents and which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice , A=1.1785(40), and D=1.0864(50), and for the FCC lattice , A=1.1736(24), and D=1.0460(50).
... For Z d it is proved in [26] that the connective constant has an asymptotic expansion to all orders in (2d) −1 , with integer coefficients, and in [11] thirteen of these coefficients are computed with the result that ...
... Also, in the asymptotic formula c n = Aµ n [1 + O(n −ǫ )] for Z d with d ≥ 5 proved in [25], the amplitude A is proved in [11] to have an asymptotic expansion to all orders, with integer coefficients, and in particular ...
... We believe that these series and also the series for the hypercube in Theorem 1.3 have radius of convergence zero but are Borel summable; to prove any of these statements is an open problem. Numerical results of Padé-Borel resummation [34] of the above series for µ(Z d ) and A(Z d ) are reported in [11,Table 15]. For the related question of the 1/d expansion for the critical point for the Berlin-Kac spherical model, it is resolved affirmatively in [21] that the radius of convergence of the expansion is zero. ...
The counting of self-avoiding walks is a classical problem in enumerative combinatorics which is also of interest in probability theory, statistical physics, and polymer chemistry. We study the number of n-step self-avoiding walks on the N-dimensional hypercube, and identify an N-dependent connective constant and amplitude such that is for all n and N, and is asymptotically as long as for any fixed . We refer to the regime as the dilute phase and regard it as the regime in which the self-avoiding walk is not yet long enough to "feel" the finite volume of the hypercube. We discuss conjectures concerning different behaviours of when n reaches and exceeds , corresponding to a critical window and a dense phase; this shares similarities with the much studied percolation phase transition on the hypercube. In addition, we prove that the connective constant has an asymptotic expansion to all orders in , with integer coefficients, and we compute the first five coefficients . A similar asymptotic expansion holds for . The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided. The generating function analysis also leads to results for the asymptotic behaviour of the susceptibility and the expected length for self-avoiding walk on the hypercube. The convergence proof we present for the lace expansion is simpler for self-avoiding walk on the hypercube than it is for other settings and models.
... In 1992, MacDonald et al. [10] reached N max = 23, and in 2000 MacDonald et al. [11] reached N max = 26. In 2007, a combination of the lace expansion and the two-step method allowed for the enumeration of SAWs up to N max = 30 steps [12]. Recently, the length-doubling method [13] was presented which allowed enumerations to be extended up to N max = 36. ...
... We reduce the influence of this additional sub-leading correction by separately treating the sequences for even and odd N . See [12] for more detailed discussion on this point for the asymptotic behaviour of Z N on the SC lattice, which is also bipartite. ...
... Differential approximants are extremely effective at extracting information about critical exponents from the long series that have been obtained for two-dimensional lattice models, such as self-avoiding polygons [22] or walks on the square lattice [23]. However, differential approximants have been far less successful for the shorter series available for three-dimensional models such as SAWs on the simple cubic lattice [12,13]. For short series, it seems that corrections-to-scaling due to confluent corrections are too strong at the orders that can be reached to be able to reliably determine critical exponents. ...
Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the exponents and which are in agreement with, but less accurate than those obtained earlier from enumeration results on the simple cubic lattice. The non-universal growth constant and amplitudes are accurately determined, yielding for the BCC lattice , A=1.1785(40), and D=1.0864(50), and for the FCC lattice , A=1.1736(24), and D=1.0460(50).
... where the constant µ and amplitude A are lattice dependent, the critical exponent γ = 1.156 953 00(95) [2] is universal, and the leading correction-to-scaling exponent ∆ 1 = 0.528(8) [3] is also universal. There are also sub-leading analytic corrections which we will not need to consider as we are working in the large N limit where these contributions are negligible compared to statistical error, but they are discussed further elsewhere, e.g. in [4] for two dimensions and [5] for three dimensions. The constant µ, variously termed the growth constant or the connective constant, is not a universal quantity, but is nonetheless of significant interest. ...
... We adopt a central estimate of 1.156955 (13), with the confidence interval chosen to capture this variability. In contrast, the bcc fits have a smooth trend and we estimate 1.156949 (5). ...
We extend a binary tree implementation of the pivot algorithm to the face-centered cubic and body-centered cubic lattices, and use it to calculate the growth constant, , for self-avoiding walks on these lattices. We find that \mufcc =10.037\, 057\, 85 \pm 0.000\, 000\, 14 and \mubcc = 6.530\, 511\, 501 \pm 0.000\, 000\, 084. In addition, we estimate the amplitudes \Afcc = 1.17119 \pm 0.00003 and \Abcc = 1.17637 \pm 0.00003, and provide convincing numerical evidence in support of the hypothesis that the critical exponent is a universal quantity.
... The computation of the expansion coefficients follows a straightforward iterative procedure and could be extended to more terms with further effort to enumerate lace graphs on the hypercube. For small lace graphs, enumeration on the hypercube is not difficult to adapt from the enumerations on Z provided in [13], and in this way we avoid any difficult counting in the computation of the five coefficients given in Theorem 1.3. ...
... Extensive computer assisted enumerations of lace graphs for Z are given in [13]. Lace graphs on the hypercube are a subset of those on Z . ...
We study the number cn(N) of n n ‐step self‐avoiding walks on the N N ‐dimensional hypercube, and identify an N N ‐dependent connective constant μN and amplitude AN such that cn(N) is O(μNn) for all n n and N N , and is asymptotically ANμNn as long as n≤2pN for any fixed p<12. We refer to the regime n≪2N/2 as the dilute phase. We discuss conjectures concerning different behaviors of cn(N) when n n reaches and exceeds 2N/2, corresponding to a critical window and a dense phase. In addition, we prove that the connective constant has an asymptotic expansion to all orders in N−1, with integer coefficients, and we compute the first five coefficients μN=N−1−N−1−4N−2−26N−3+O(N−4). The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.
... The statistics of SAWs are used to model the physics of polymer chains with excluded volume interactions [11]. The universal quantities of SAWs, such as the growth exponent, size-scaling exponent, and finite-length corrections can be calculated by full enumerations of SAWs [3,8] or by stochastic generation of large ensembles [2] on various lattices. These universal SAW properties make predictions for the behavior of real polymers that can be observed in experiment, for example the dependence of radius of gyration on molecular weight [4]. ...
... These universal SAW properties make predictions for the behavior of real polymers that can be observed in experiment, for example the dependence of radius of gyration on molecular weight [4]. Combinatorial techniques used to enumerate SAWs include the lace expansion method [3], transfer matrix methods [8], and the length-doubling method [14]. ...
A growing self-avoiding walk (GSAW) is a stochastic process that starts from the origin on a lattice and grows by occupying an unoccupied adjacent lattice site at random. A sufficiently long GSAW will reach a state in which all adjacent sites are already occupied by the walk and become trapped, terminating the process. It is known empirically from simulations that on a square lattice, this occurs after a mean of 71 steps. In Part I of a two-part series of manuscripts, we consider simplified lattice geometries only two sites high ("ladders") and derive generating functions for the probability distribution of GSAW trapping. We prove that a self-trapping walk on a square ladder will become trapped after a mean of 17 steps, while on a triangular ladder trapping will occur after a mean of 941/48 (~19.6 steps). We discuss additional implications of our results for understanding trapping in the "infinite" GSAW.
... The correction terms a i /N i are called analytical and a Δ i /N Δ+i (here, Δ is not an integer) are called non-analytic correction terms. For fully flexible SAW model on the simple cubic lattice the recent result Δ = 0.528 (8), estimated by high-resolution Monte Carlo simulations of SAWs [28], demonstrates that for three-dimensional SAWs the leading correction-to-scaling term is non-analytic [29]. We notice that this finding for exponent Δ is consistent with the value 0.5310(33) obtained by renormalization group calculation within the Domb-Joyce model [10]. ...
... where the exponent Δ = 0.528(8) governs non-analytical correction terms, for SAWs in three-dimensional space. To learn quantities γ and μ, we examine the following ratio [29] Z N +1 ...
... The length of the path, ℓ, is given by the number of steps made until the walk is terminated. A large number of studies of SAWs on regular lattices were devoted to the enumeration of paths as a function of their length [26][27][28][29][30][31]. These studies provided much insight on the structure and thermodynamics of polymers [6,7]. ...
... the initial node is not isolated). Thus, this tail distribution takes While equation(30) applies to any network, in the case of ER networks we have an explicit expression for the probability of a node to become isolated, ...
We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erd\H{o}s-R\'enyi networks we show that the pruned networks maintain a Poisson degree distribution, , with an average degree, , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, , increases dramatically as a function of . We also obtain analytical results for the path-length distribution, , of the SAW paths which are actually pursued, starting from a random initial node. It turns out that follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.
... From our series, and known series for 3d SAWs [8,31], we have calculated K n for n ≤ 27 using exact coefficients, and for n ≤ 34 using the last seven approximate coefficients. From the expected asymptotic form of the coefficients, we expect ...
... The three innovations are the use of the lace expansion and the two-step method [8], and the length-doubling algorithm [31]. ...
We study terminally attached self-avoiding walks and bridges on the
simple-cubic lattice, both by series analysis and Monte Carlo methods. We
provide strong numerical evidence supporting a scaling relation between
self-avoiding walks, bridges, and terminally attached self-avoiding walks, and
posit that a corresponding amplitude ratio is a universal quantity.
... Good numerical estimates for µ can be found in the literature, but the entropic exponent α is known a lesser accuracy. A good estimate for µ was obtained in [4]: µ = 4.684044 (11), (1.4) α ≈ 0.24. (1.5) ...
... After p nmin (K) is estimated, the flatGAS data can be used to estimate p n (K) using the ratios of collected weights as in Eq. (3.6). Some data for the unknot, the trefoil and the figure eight knot are listed in Table 2. Observe that series for all polygons is known to n = 32 [4] and that a comparison (for n < 24 before the first trefoils appear) shows that flatGAS gives good estimates: For example, p 18 (0 1 ) = 108088232 and p 22 (0 1 ) = 29764630632, close to the estimates in Table 2. ...
Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and may be used as a model of knotting in ring polymers. In this paper we study the effects of knotting on the conformational entropy of lattice polygons and so determine the relative fraction of polygons of different knot types at large lengths. More precisely, we consider the number of cubic lattice polygons of n edges with knot type K, pn(K). Numerical evidence strongly suggests that as n → ∞, where μ0 is the growth constant of unknotted lattice polygons, α is the entropic exponent of lattice polygons, and NK is the number of prime knot components in the knot type K (see the paper [Asymptotics of knotted lattice polygons, J. Phys. A: Math. Gen.31 (1998) 5953–5967]). Determining the exact value of pn(K) is far beyond current techniques for all but very small values of n. Instead we use the GAS algorithm (see the paper [Generalised atmospheric sampling of self-avoiding walks, J. Phys. A: Math. Theor.42 (2009) 335001–335030]) to enumerate pn (K) approximately. We then extrapolate ratios [pn(K)/pn(L)] to larger values of n for a number of given knot types. We give evidence that for the unknot 01 and the trefoil knot 31, there exists a number M01, 31 ≈170000 such that pn (01) > pn (31) if n < M01, 31 and pn (01) ≤pn (31) if n ≥M01, 31. In addition, the asymptotic relative frequencies for a variety of knot types are determined. For example, we find that [pn(31)/pn(41)] → 27.0 ± 2.2, implying that there are approximately 27 polygons of the trefoil knot type for every polygon of knot of type 41 (the figure eight knot), in the asymptotic limit. Finally, we examine the dominant knot types at moderate values of n and conjecture that the most frequent knot types in polygons of any given length n are of the form (or its chiral partner), where are right- and left-handed trefoils, and N increases with n.
... For bipartite lattices there is an additional "anti-ferromagnetic" term which has a factor of (−1) N . It is important to take this into account when studying series from exact enumeration [2], but it is negligible for the values of N that are accessible to the Monte Carlo computer experiments considered here and so we neglect it. In two dimensions the critical exponent γ is known exactly, predicted to be 43/32 over thirty years ago via Coulomb gas arguments by Nienhuis [3]. ...
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent for self-avoiding walks to unprecedented accuracy. Our final estimate for is 1.15695300(95).
... It can also be regarded as a test-bed problem for physically relevant models which have logarithmic corrections to scaling, for example the θ-transition in three dimensions is believed to be identified with a tricritical point, and has meanfield behavior with logarithmic corrections. Four-dimensional SAWs have been studied by Monte Carlo [3,4] and enumeration [5][6][7] methods, and we note that rigorous results have recently been obtained for the 4-dimensional weakly self-avoiding walk via the rigorous renormalization group [8]. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... We shifted the value of N of Eq. 1 by an amount δN x to obtain smoother convergence by altering the sub-leading corrections (see e.g. [15]); estimates for ν, ∆ 1 , D x , and b x are unaffected in the limit N min → ∞. ...
We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to steps. Consequently the critical exponent for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is . The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.
... Each node consists of two AMD Barcelona 2.3GHz quad core processors. In principle, it is probably possible to obtain another term or two by applying the two-step method [14], but we have not pursued this here. ...
We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O for a series of length n. We have extended the definition to three dimensions and produced series expansions for both prudent walks and polygons in three dimensions. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct classes of polygons, we find that the growth constant for polygons varies with class, while for walks it does not. We give the critical exponent for both walks and polygons. In the three-dimensional case we estimate the growth constant for both walks and polygons and also estimate the usual critical exponents and $\alpha.
... A. Self-avoiding walks (n = 0) Over the last decade, successive improvements of Monte Carlo methods significantly diminished the uncertainty of critical exponents [52][53][54][55]. The latest and most accurate estimates are γ = 1.156953(1) [53] and ν = 0.5875970(4) [54]. ...
We present the perturbative renormalization group functions of O(n)-symmetric theory in dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.
... A self-avoiding walk (SAW) is a walk on a graph that does not visit the same vertex twice [MS13]. Self-avoiding walks are used to model polymer chains in good solvent conditions, and exact enumerations of SAWs provide insight into scaling exponents that can be measured in polymer experiments [CLS07]. Typically, each SAW of a given length is treated as equally probable such that the average size of a polymer chain, which is able to adopt these equally likely configurations through thermal fluctuations, may be computed. ...
A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and has a trapped endpoint. We show that the generating function enumerating GSAWs on a half-infinite strip of finite height is rational, and we give a procedure to construct a combinatorial finite state machine that allows one to compute this generating function. We then modify this procedure to compute generating functions for GSAWs under two probabilistic models. We perform Monte Carlo simulations to estimate the expected length and displacement for GSAWs on the quarter plane, half plane, full plane, and half-infinite strips of bounded height for which we cannot compute the generating function. Finally, we prove that the generating functions for Greek key tours (GSAWs on a finite grid that visit every vertex) on a half-infinite strip of fixed height are also rational, allowing us to resolve several conjectures.
... The existence of A has been derived using the lace expansion for the self-avoiding walk model, see for instance[CLS07]. To the best of our knowledge, the exact asymptotic of the susceptibility is not known for Bernoulli percolation. ...
We consider the critical FK-Ising measure on with . We construct the measure and prove it satisfies . This corresponds to the natural candidate for the incipient infinite cluster measure of the FK-Ising model. Our proof uses a result of Lupu and Werner (Electron. Commun. Probab., 2016) that relates the FK-Ising model to the random current representation of the Ising model, together with a mixing property of random currents recently established by Aizenman and Duminil-Copin (Ann. Math., 2021). We then study the susceptibility of the nearest-neighbour Ising model on . When , we improve a previous result of Aizenman (Comm. Math. Phys., 1982) to obtain the existence of such that, for , \begin{equation*} \chi(\beta)= \frac{A}{1-\beta/\beta_c}(1+o(1)), \end{equation*} where o(1) tends to 0 as tends to . Additionally, we relate the constant A to the incipient infinite cluster of the double random current.
... The critical exponents are considered universal, while the values of A, D, and µ are lattice dependent. Starting from the work of Orr [22], elegant methods and efficient algorithms have been developed over the years to tackle the exponentially difficult SAW enumeration problem [6,9,36,38,39,61,62,89,90], reaching various high-N SAWs in distinct lattices. We should note here that our SAW enumeration algorithm is not as efficient as the state-of-the-art methods described above, and thus our analysis is limited to SAWs of intermediate number of steps. ...
Recent simulation studies have revealed a wealth of distinct crystal polymorphs encountered in the self-organization of polymer systems driven by entropy or free energy. The present analysis, based on the concept of self-avoiding random walks (SAWs) on crystal lattices, is useful to calculate upper bounds for the entropy difference of the crystals that are formed during polymer crystallization and thus to predict the thermodynamic stability of distinct polymorphs. Here, we compare two pairs of crystals sharing the same coordination number, ncoord: hexagonal close-packed (HCP) and face centered cubic (FCC), both having ncoord = 12 and the same packing density, and the less dense simple hexagonal (HEX) and body centered cubic (BCC) lattices, with ncoord = 8. In both cases, once a critical number of steps is reached, one of the crystals shows a higher number of SAWs compatible with its geometry. We explain the observed trends in terms of the bending and torsion angles as imposed by the geometric constraints of the crystal lattice.
... All reported numerical calculations have been executed on an Intel i9-10850K with 16 Gb of memory, running on Linux operating system. Starting from the work of Orr [22] elegant methods and efficient algorithms have been developed over the years to tackle the exponentially difficult SAW enumeration problem [6,9,36,38,39,61,62,88,89] allowing to reach various high-N SAWs in distinct lattices. We should further note that our SAW enumeration algorithm is not as efficient as the state-of-the-art methods described above, and thus our analysis is limited to SAWs of intermediate number of steps. ...
Recent simulation studies have revealed a wealth of distinct crystal polymorphs encountered in the self-organization of polymer systems driven by entropy or free energy. The present analysis, based on the concept of self-avoiding random walks on crystal lattices, is useful to calculate upper bounds for the entropy difference of the crystals that are formed during polymer crystallization and thus provide predictions on polymorph thermodynamic stability. Here, we compare two pairs of crystals sharing the same coordination number, ncoord: hexagonal close packed (HCP) and face centered cubic (FCC), both having ncoord = 12 and the same packing density, and the less dense hexagonal (HEX) and body centered cubic (BCC) lattices, with ncoord = 8. In both cases, once a critical step length is reached, one of the crystals shows a higher number of SAWs compatible with the crystal. We explain the observed trends in terms of the bending and torsion angles corresponding to the different chain geometry as imposed by the geometric constraints of the crystal lattice.
... In view of the reversibility of the moves, this is equivalent to the capability of reaching a specific target (say, e.g., a 4-step SAW completely straighten along the x direction) from any n-step SAW. It is easy to figure out a possible finite sequence of moves performing the last task as follows: first, the application of n−4 moves of the kind −1 reaches one of the 726 [53,54] distinct 4-step SAWs occurring on the cubic lattice; then, within a finite sequence of n-preserving moves the target is certainly reached. In all our simulations for the motion of CM we start from a grand canonical equilibrium configuration. ...
Three-dimensional Monte Carlo simulations provide a striking confirmation to a recent theoretical prediction: the Brownian non-Gaussian diffusion of critical self-avoiding walks. Although the mean square displacement of the polymer center of mass grows linearly with time (Brownian behavior), the initial probability density function is strongly non-Gaussian and crosses over to Gaussianity only at large time. Full agreement between theory and simulations is achieved without the employment of fitting parameters. We discuss simulation techniques potentially capable of addressing the study of anomalous diffusion under complex conditions like adsorption- or Theta-transition.
... In view of the reversibility of the moves, this is equivalent to the capability of reaching a specific target (say, e.g., a four-step SAW completely straighten along the x direction) from any n-step SAW. It is easy to figure out a possible finite sequence of moves performing the last task as follows: first, the application of n − 4 moves of the kind −1 reaches one of the 726 [53,54] distinct four-step SAWs occurring on the cubic lattice; then, within a finite sequence of n-preserving moves the target is certainly reached. In all our simulations for the motion of CM we start from a grand canonical equilibrium configuration. ...
Three-dimensional Monte Carlo simulations provide a striking confirmation to a recent theoretical prediction: the Brownian non-Gaussian diffusion of critical self-avoiding walks. Although the mean square displacement of the polymer center of mass grows linearly with time (Brownian behavior), the initial probability density function is strongly non-Gaussian and crosses over to Gaussianity only at large time. Full agreement between theory and simulations is achieved without the employment of fitting parameters. We discuss simulation techniques potentially capable of addressing the study of anomalous diffusion under complex conditions like adsorption- or Theta-transition.
... One applies γ ≡ γ 1 ≈ 0.678 [61][62][63] for strands having one end grafted, and γ ≡ γ 11 ≈ −0.39 62,63 for strands having both ends surface attached. The partition sum in free solution scales as Z free $ μ N N γ 0 À1 with γ 0 ≈ 1.1567 [64][65][66] . Since we further compare only ratios of partition sums for given total chain length, we assume that q-and μ-dependent contributions cancel up to a factor of the order unity. ...
Recent developments in computer processing power lead to new paradigms of how problems in many-body physics and especially polymer physics can be addressed. Parallel processors can be exploited to generate millions of molecular configurations in complex environments at a second, and concomitant free-energy landscapes can be estimated. Databases that are complete in terms of polymer sequences and architecture form a powerful training basis for cross-checking and verifying machine learning-based models. We employ an exhaustive enumeration of polymer sequence space to benchmark the prediction made by a neural network. In our example, we consider the translocation time of a copolymer through a lipid membrane as a function of its sequence of hydrophilic and hydrophobic units. First, we demonstrate that massively parallel Rosenbluth sampling for all possible sequences of a polymer allows for meaningful dynamic interpretation in terms of the mean first escape times through the membrane. Second, we train a multi-layer neural network on logarithmic translocation times and show by the reduction of the training set to a narrow window of translocation times that the neural network develops an internal representation of the physical rules for sequence-controlled diffusion barriers. Based on the narrow training set, the network result approximates the order of magnitude of translocation times in a window that is several orders of magnitude wider than the training window. We investigate how prediction accuracy depends on the distance of unexplored sequences from the training window.
... b Using Eqs. (74) and (75) of Ref.[22] with 0.516 ≤ ∆ 1 ≤ 0.54. c No error estimates were made in Ref.[24], but estimates for ν were in the range 0.5870 ≤ ν ≤ 0.5881. ...
The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to monomers, we find that the ratio takes the value , which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general, and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of .
... They have C = √ 2µ on any lattice, which for the cubic lattice gives C ≈ 3.06 and allows them to count SAWs of length up to 36. The best count of SAPs for the cubic lattice is in [2], which goes up to length 32. ...
We give an algorithm for counting self-avoiding walks or self-avoiding polygons of length n that runs in time on 2-dimensional lattices and time on d-dimensional lattices for d > 2.
... Kesten [37] proved the asymptotic expansion µ w = 2d − 1 − 1/2d + O(1/d 2 ) using finite memory walks. Since then, several terms of the asymptotic expansion have been computed using the lace expansion (see for example [6,32]). ...
We prove that for the d-regular tessellations of the hyperbolic plane by k-gons, there are exponentially more self-avoiding walks of length n than there are self-avoiding polygons of length n, and we deduce that the self-avoiding walk is ballistic. The latter implication is proved to hold for arbitrary transitive graphs. Moreover, for every fixed k, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion as ; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length n is comparable to the nth power of their connective constant. Some of these results were previously obtained by Madras and Wu \cite{MaWuSAW} for all but finitely many regular tessellations of the hyperbolic plane.
... Good numerical estimates and rigorous bounds on the connective constant are known, but the exact value for Z d is not known for any d ≥ 2. For SAWs defined instead on the hexagonal lattice, it has been proved that μ = 2 + √ 2 [2]. As the dimension d goes to infinity, there is an asymptotic expansion μ ∼ 2d − 1 + ∞ n=1 a n (2d) −n with integer coefficients a n whose values are known up to and including a 11 [3]. The connective constant for SAWs in more general settings than Z d is a topic of current research [4]. ...
The self-avoiding walk, and lattice spin systems such as the φ⁴ model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous non-perturbative renormalization group method.
... For simple non-SAWs, γ = 1 and ν = 1 2 . Estimates and bounds for µ, ν and γ for SAWs are available [25,[83][84][85][86][87][88][89]. Approximate values in three dimensions are µ ≈ 4.684, γ ≈ 1.157 and ν = 0.588. ...
Polymers in highly confined geometries can display complex morphologies including ordered phases. A basic component of a theoretical analysis of their phase behavior in confined geometries is the knowledge of the number of possible single-chain conformations compatible with the geometrical restrictions and the established crystalline morphology. While the statistical properties of unrestricted self-avoiding random walks (SAWs) both on and off-lattice are very well known, the same is not true for SAWs in confined geometries. The purpose of this contribution is (a) to enumerate the number of SAWs on the simple cubic (SC) and face-centered cubic (FCC) lattices under confinement for moderate SAW lengths, and (b) to obtain an approximate expression for their behavior as a function of chain length, type of lattice, and degree of confinement. This information is an essential requirement for the understanding and prediction of entropy-driven phase transitions of model polymer chains under confinement. In addition, a simple geometric argument is presented that explains, to first order, the dependence of the number of restricted SAWs on the type of SAW origin.
... Good numerical estimates and rigorous bounds on the connective constant are known, but the exact value for Z d is not known for any d ≥ 2. For SAWs defined instead on the hexagonal lattice, it has been proved that µ = 2 + √ 2 [25]. As the dimension d goes to infinity, there is an asymptotic expansion µ ∼ 2d − 1 + ∞ n=1 a n (2d) −n with integer coefficients a n whose values are known up to and including a 11 [22]. The connective constant for SAWs in more general settings than Z d is a topic of current research [35]. ...
The self-avoiding walk, and lattice spin systems such as the model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous nonperturbative renormalisation group method.
... A. Self-avoiding walks (n = 0) Over the last decade, successive improvements of Monte Carlo methods significantly diminished the uncertainty of critical exponents [51][52][53][54]. The latest and most accurate estimates are γ = 1.156953(1) [52] and ν = 0.5875970(4) [53]. ...
We present the perturbative renormalization group functions of O(n)-symmetric theory in dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.
... Four-dimensional SAWs have been studied by Monte Carlo [14,24] B Nathan Clisby nclisby@swin.edu.au 1 Department of Mathematics, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia and enumeration [6,11,19] methods, and we note that rigorous results have recently been obtained for the 4-dimensional continuous-time weakly self-avoiding walk via the rigorous renormalization group [2][3][4]. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... For bipartite lattices there is an additional 'anti-ferromagnetic' term which has a factor of (−1) N . It is important to take this into account when studying series from exact enumeration [2], but it is negligible for the values of N that are accessible to the Monte Carlo computer experiments considered here and so we neglect it. ...
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent for self-avoiding walks to unprecedented accuracy. Our final estimate for is 1.15695300(95).
... This is a random walk which does not visit the same node more than once [23]. At each time step, the walker chooses its next move randomly from the neighbors of its present node, excluding nodes which were already visited [24][25][26][27][28][29][30]. The path terminates when the SAW reaches a stalemate situation, namely a dead end node which does not have any yet unvisited neighbors. ...
We present analytical results for the distribution of first hitting times of random walkers on Erd\H{o}s-R\'enyi networks. Starting from a random initial node, a random walker hops randomly between adjacent nodes on the network until it hits a node which it has already visited before. At this point, the path is terminated. The path length d, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, . The results are found to be in excellent agreement with simulations. It turns out that the distribution follows a product of an exponential distribution and a Rayleigh distribution. We also obtain expressions for the mean, median and standard deviation of this distribution in terms of the network size and its mean degree. It is found that the first hitting time is much shorter than the last hitting time of the corresponding self-avoiding walk. The termination of the path may take place either due to a backtracking step of the random walker into the previous node or due to retracing of its path, namely stepping into a node which has been visited two or more time steps earlier. We obtain analytical results for the probabilities, and , that the cause of termination will be backtracking or retracing, respectively. It is shown that in dilute networks the dominant termination scenario is backtracking while in dense networks most paths are terminated by retracing. We also obtain expressions for the conditional distributions of path lengths, and . These results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.
... which is the desired power law. Finally, using the RG derived values for ν = 0.5876 and γ = 1.1568 7/6, [23,34,35] we obtain ...
We compute the effects of excluded volume on the probability for double-stranded DNA to form a loop. We utilize a Monte Carlo algorithm for generation of large ensembles of self-avoiding wormlike chains, which are used to compute the J factor for varying length scales. In the entropic regime, we confirm the scaling-theory prediction of a power-law drop off of -1.92, which is significantly stronger than the -1.5 power law predicted by the non-self-avoiding wormlike chain model. In the elastic regime, we find that the angle-independent end-to-end chain distribution is highly anisotropic. This anisotropy, combined with the excluded volume constraints, leads to an increase in the J factor of the self-avoiding wormlike chain by about half an order of magnitude relative to its non-self-avoiding counterpart. This increase could partially explain the anomalous results of recent cyclization experiments, in which short dsDNA molecules were found to have an increased propensity to form a loop.
... where α is some constant, and the correction term θ ≈ .5 is obtained from the asymptotic form [5]. We obtain the following results for d = 2, 3 in table 3.4. ...
... Restricting model structure space to maximally compact conformations in 2D [306] or 3D [307,308] reduces computation drastically because such conformations constitute only a tiny fraction of all possible conformations [280,[309][310][311][312]. For instance, whereas the total number of all distinguishable conformations (not related by rotations and reflections) for a chain with 25 residues configured on the 2D square lattice is 5 768 299 665 [313], the number of maximally compact 25-residue conformations restricted to a 5 Â 5 square, as considered in the study of Taverna & Goldstein [306], is only 1081 [309]. In 3D, the total number of all possible conformations (not related by rotations and inversions) on the simple cubic lattice for a chain with 27 residues is 11 447 808 041 780 409 [313,314], but the number of maximally compact 27-residue conformations restricted to a 3 Â 3 Â 3 cube, as considered by Deeds & Shakhnovich [307], is only 103 346 [280]. ...
The study of molecular evolution at the level of protein-coding genes often entails comparing large datasets of sequences to infer their evolutionary relationships. Despite the importance of a protein's structure and conformational dynamics to its function and thus its fitness, common phylogenetic methods embody minimal biophysical knowledge of proteins. To underscore the biophysical constraints on natural selection, we survey effects of protein mutations, highlighting the physical basis for marginal stability of natural globular proteins and how requirement for kinetic stability and avoidance of misfolding and misinteractions might have affected protein evolution. The biophysical underpinnings of these effects have been addressed by models with an explicit coarse-grained spatial representation of the polypeptide chain. Sequence–structure mappings based on such models are powerful conceptual tools that rationalize mutational robustness, evolvability, epistasis, promiscuous function performed by ‘hidden’ conformational states, resolution of adaptive conflicts and conformational switches in the evolution from one protein fold to another. Recently, protein biophysics has been applied to derive more accurate evolutionary accounts of sequence data. Methods have also been developed to exploit sequence-based evolutionary information to predict biophysical behaviours of proteins. The success of these approaches demonstrates a deep synergy between the fields of protein biophysics and protein evolution.
... which is the desired power law. Finally, using the RG derived values for ν = 0.5876 and γ = 1.1568 7/6, [23,34,35] we obtain ...
We compute for the first time the effects of excluded volume on the
probability for double-stranded DNA to form a loop. We utilize a Monte-Carlo
algorithm for generation of large ensembles of self- avoiding worm-like chains,
which are used to compute the J-factor for varying lengthscales. In the
entropic regime, we confirm the scaling-theory prediction of a power-law drop
off of -1.92, which is significantly stronger than the -1.5 power-law predicted
by the non-self-avoiding worm-like chain model. In the elastic regime, we find
that the angle-independent end-to-end chain distribution is highly anisotropic.
This anisotropy, combined with the excluded volume constraints, lead to an
increase in the J-factor of the self-avoiding worm-like chain by about half an
order of magnitude relative to its non-self-avoiding counterpart. This increase
could partially explain the anomalous results of recent cyclization
experiments, in which short dsDNA molecules were found to have an increased
propensity to form a loop.
We prove that for the ‐regular tessellations of the hyperbolic plane by ‐gons, there are exponentially more self‐avoiding walks of length than there are self‐avoiding polygons of length . We then prove that this property implies that the self‐avoiding walk is ballistic, even on an arbitrary vertex‐transitive graph. Moreover, for every fixed , we show that the connective constant for self‐avoiding walks satisfies the asymptotic expansion as ; on the other hand, the connective constant for self‐avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self‐avoiding walks of length is comparable to the th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.
The negative internal energetic contribution to the elastic modulus (negative energetic elasticity) has been recently observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the n-step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We theoretically demonstrate the emergence of negative energetic elasticity based on an exact enumeration up to n=20 and analytic expressions for arbitrary n in special cases. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer–solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.
The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as zc=0.147622380(2), improving over the existing result zc=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near zc, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L2(lnL)2ŷh and C∼(lnL)2ŷt, and the logarithmic exponents are determined as ŷh=0.251(2) and ŷt=0.25(3), in excellent agreement with the field theoretical prediction ŷh=ŷt=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.
Series expansions remain, in many cases, one of the most accurate ways of estimating critical exponents. Historically it was the results from series expansions that suggested universality at criticality. Two expansions will be considered in this chapter. In the high-temperature series, the Boltzmann factor is expanded in powers of the inverse temperature, and the sum over all configurations is taken term by term. In the Ising model, this leads to an expansion in powers of tanh(J∕T)≪1. In the low-temperature expansion, configurations are counted in order of their importance as the temperature is increased from zero. Starting from the ground state, the series is constructed by successively adding terms from 1, 2, 3, … flipped spins. In the Ising model, this leads to an expansion in powers of exp(−2J∕T)≪1.
Polymers in highly confined geometries can display complex morphologies including ordered phases. A basic component of a theoretical analysis of their phase behavior in confined geometries is the knowledge of the number of possible single-chain conformations compatible with the geometrical restrictions and the established crystalline morphology. While the statistical properties of unrestricted self-avoiding random walks (SAWs) both on and off-lattice are very well known, the same is not true for SAWs in confined geometries. The purpose of this contribution is a) to enumerate the number of SAWs on the simple cubic (SC) and face-centered cubic (FCC) lattices under confinement for moderate SAW lengths, and b) to obtain an approximate expression for their behavior as a function of chain length, type of lattice, and degree of confinement. This information is an essential requirement for the understanding and prediction of entropy-driven phase transitions of model polymer chains under confinement. In addition, a simple geometric argument is presented that explains, to first order, the dependence of the number of restricted SAWs on the type of SAW origin.
The aim of these notes is to provide an introduction to the topic of statistical topology. With this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments. Some questions that we are going to address are the following: (i) What is a knot and how can we identify it? (ii) Which is the probability of finding a random curve that is knotted? (iii) How complex are these knots? (iv) How big are they? To try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some "state of the art" theoretical and numerical techniques. Exercises and lists of open problems will be provided too.
The universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self-avoiding walks is estimated by high-resolution Monte Carlo simulations. By studying chains of length of up to N=225≈34×106 monomers, we find that the ratio takes the value RG/RH=1.5803940(45), which is several orders of magnitude more accurate than the previous state of the art. This is facilitated by a sampling scheme which is quite general and which allows for the efficient estimation of averages of a large class of observables. The competing corrections to scaling for the hydrodynamic radius are clearly discernible. We also find improved estimates for other universal properties that measure the chain dimension. In particular, a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the Flory exponent of ν=0.58759700(40).
Polymers are important macromolecules in many physical, chemical, biological and industrial problems. Studies on simple lattice polymer models are very helpful for understanding behaviors of polymers. We develop an efficient algorithm for computing exact partition functions of lattice polymer models, and we use this algorithm and personal computers to obtain exact partition functions of the interacting self-avoiding walks with monomers on the simple cubic lattice up to and on the square lattice up to . Our algorithm can be extended to study other lattice polymer models, such as the HP model for protein folding and the charged HP model for protein aggregation. It also provides references for checking accuracy of numerical partition functions obtained by simulations.
We develop a new approach for scaling analysis of N-step Random Walks
(RWs). The mean square end-to-end distance, , is
written as ensemble averages of dot products between the walker's position and
displacement vectors at the j-th step, which we call inner persistence
lengths (IPLs). We analytically introduce a novel relation between
and the classical persistence length for
polymers, , for RW models statistically invariant under orthogonal
transformations. With Monte Carlo simulations for 2D and 3D lattices, we find
that converges to a constant apart from scaling corrections.
Building SAWs with typically one hundred steps, we accurately estimate
exponents, and , from IPL
behavior as function of j. The obtained results are in excellent agreement
with those in the literature, providing strong evidence that the whole
information about the scaling behavior of is
into only one ensemble of same length paths.
Series expansions remain, in many cases, one of the most accurate ways of estimating critical exponents. The high-temperature expansion is an expansion in powers of the inverse temperature. In the low-temperature expansion configurations are counted in order of their importance as the temperature is increased from zero. Starting from the ground state the series is constructed by successively adding terms with and increasing number of flipped spins. Each term in the high- or low-temperature series is represented by a graph on a lattice and constructing the series amounts to counting the graphs belonging to a fixed order in the expansions. With various extrapolations, for example with Padé approximants, we extract the behavior of a lattice system near a critical point and in particular its critical exponents. The expansions are calculated to high orders for the Ising model in two and three dimensions and for the non-linear O(N) lattice model. At the end we consider polymers and self-avoiding walks. In several tables we compare predictions for the critical temperatures and critical exponents of various expansions, renormalization group techniques and lattice simulations.
The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1} m obtained in the series of papers (Borgs et al. in Random Struct Algorithms 27:137–184, 2005; Borgs et al. in Ann Probab 33:1886–1944, 2005; Borgs et al. in Combinatorica 26:395–410, 2006; van der Hofstad and Nachmias in Hypercube percolation, Preprint 2012). Secondly, we explain how this study can be performed without the use of the so-called “lace expansion” technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability.
In this work the excluded volume problem of a linear flexible polymer molecule in a solution was investigated using a new method. The Domb-Joyce (DJ) lattice model (Domb C. and Joyce G. S. (1972). J. Phys. C: Solid State Phys. 5 956) was used to describe the polymer chain with a varying excluded volume parameterramateur w and bond number N. Monte Carlo (MC) generated data for the mean square end-to-end distance Rsbsp{N}{2} and the second virial coefficient Asb{2,N} were analyzed by a renormalization group technique that is a generalization of the one-parameter recursion model (Nickel B. G. (1991). Macromolecules 24, 1358). By defining the effective exponents nusb{R}(N,psi) and nusb{A}(N,psi ) using 2sp{2nusb{R}} = Rsbsp{2N}{2}/Rsbsp{N}{2} and 2sp{3nusb{A}} = Asb{2,2N}/Asb{2,N} where psi = {1/4}({6/pi})sp{3/2}{{Asb{2,N}}/{Rsbsp{N}{3}}} is the interpenetration function, the corrections varying as Nsp{-Delta} were eliminated from nusb{R}(N,psi) and nusb{A}(N,psi) and both universal critical exponents nu and Delta of the expected long chain behaviors Rsbsp{N}{2}~ asb{R}Nsp{2nu}(1 + bsb{R}Nsp{-Delta} +\...) and Asb{2,N}~ asb{A}Nsp{3nu}(1 + bsb{A}Nsp{-Delta} +\...) were determined very accurately. The problems encountered by standard methods when extracting the values of the leading exponent nu and the correction to scaling exponent Delta from the finite chain data were eliminated by the simultaneous use of many models (i.e., w in the range of 0 < omega 0 and N->infty with z~ wNsp{1/2} = const.) our result for the expansion factor alphasbsp{R}{2}(z) agrees very well with the previous high precision estimate of des Cloizeaux et al. (des Cloizeaux J., Conte R. and Jannik G. (1985). Journal de Physique Lettres 46, L-595) in the range z
High temperature series expansion for the critical exponents of the Ising model are reanalyzed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent gamma in three dimensions, close to 1. 245, therefore lower than the quoted value 1. 250, and much closer to the renormalization group (R. G. ) value 1. 241. The exponent is analyzed in two ways: in one method gamma is estimated directly while in the other one T//c, is calculated first. With these new values of T//c, the exponent alpha is recalculated and found to be very close to the R. G. value 0. 110. The value of v is not modified (0. 638) and is therefore still a problem for hyperscaling, and is in disagreement with the R. G. value (0. 630).
We examine self-avoiding walks in dimensions 4 to 8 using high-precision Monte Carlo simulations up to length N = 16 384, providing the first such results in dimensions d > 4 on which we concentrate our analysis. We analyse the scaling behaviour of the partition function and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and compare our results to 1/d-expansions and to excellent rigorous bounds that exist. In particular, we obtain precise values for the connective constants, µ5 = 8.838 544(3), µ6 = 10.878 094(4), µ7 = 12.902 817(3), µ8 = 14.919 257(2) and give a revised estimate of µ4 = 6.774 043(5). All of these are by at least one order of magnitude more accurate than those previously given (from other approaches in d > 4 and all approaches in d = 4). Our results are consistent with most theoretical predictions, though in d = 5 we find clear evidence of anomalous N -1/2-corrections for the scaling of the geometric size of the walks, which we understand as a non-analytic correction to scaling of the general form N (4-d)/2 (not present in pure Gaussian random walks).
Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields and ) of the three-dimensional field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N-vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined.
The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values.
Finally, because an error has been discovered in the last order of the published expansions (order ), we have also re-analysed the determination of exponents from the -expansion.
The conclusion is that the general agreement between -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.
We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice
d
. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for highd, and in fact agree with the first four terms of the 1/d expansion for the connective constant. The bounds are the best to date for dimensionsd 3, but do not produce good results in two dimensions. Ford=3, 4, 5, and 6, respectively, our lower bound is within 2.4%, 0.43%, 0.12%, and 0.044% of the value estimated by series extrapolation.
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 enumeration data. Our bounds are best for high d, and in fact agree with the rst four terms of the 1=d expansion for the connective constant. The bounds are the best to date for dimensions d 3, but do not produce good results in two dimensions. For d = 3; 4; 5; 6, respectively, our lower bound is within 2:4%, 0:43%, 0:12%, 0:044% of the value estimated by series extrapolation.
We prove existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on Z d . For the critical point, defined to be the reciprocal of the connective constant, the coefficients of the expansion are computed through order d 06 , with a rigorous error bound of order d 07 . Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on Z d gives the 1=d-expansion for the critical point through order d 03 , with a rigorous error bound of order d 04 . The method uses the lace expansion. 1 Introduction Expansions in the inverse dimension, or 1=d-expansions, are ubiquitous in the physics literature and serve to give approximate information in situations where more precise results are unavailable. They provide useful comparisons for other approximate methods, as well as evidence which can be used in support of or against conjecture...
Let and denote the critical values for nearest-neighbour bond percolation on the n-cube and on , respectively. Let for and for denote the degree of . We use the lace expansion to prove that for both and , p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}). This extends by two terms the result p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2}) of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for .
Nickel has generated 21 term series for the susceptibility and the correlation length on the B.C.C. lattice, for various values of the spin. We analyse them here using a variant of the ratio method. Our results fully support Nickel's conclusions i.e. that is no apparent disagreement any longer for the exponents γ and ν between the renormalization group and high temperature series estimates. We believe that these results shed serious doubts about any claim, based on shorter series, of evidence for hyperscaling violations. Nickel a calculé récemment les développements à haute température de la susceptibilité magnétique, et de la longueur de corrélation du modèle d'Ising de spin arbitraire, jusqu'à l'ordre 21. Nous avons analysé ces développements par une variante de la méthode des rapports. Nos résultats sont en très bon accord avec ceux de Nickel, et montrent qu'il n'y a plus de différence significative entre les valeurs des exposants γ et ν obtenus à partir du groupe de renormalisation et des séries de haute température. Nous croyons que ces nouveaux résultats jettent un doute sur des résultats basés sur des séries plus courtes et concluant à la violation de 1'<< hyperscaling >>.
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is the connective constant (i.e. γ=1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forcn(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (μ−−Z)1/2. The critical two-point function is shown to decay at least as fast as ⋎x⋎−2, and its Fourier transform is shown to be asymptotic to a multiple ofk−2 ask→0 (i.e. η=0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.
25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining gamma=1.2373(2), nu=0.63012(16), alpha=0.1096(5), eta=0.036 39(15), beta=0.326 53(10), and delta=4.78 93(8). Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate Delta=0.52(3) for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.
Single three-dimensional polymers confined to a slab, i.e., to the region between two parallel plane walls, are studied by Monte Carlo simulations. They are described by N-step walks on a simple cubic lattice confined to the region 1< or = z < or = D. The simulations cover both regions D RF (where RF approximately Nnu is the Flory radius, with nu approximately 0.587), as well as the cross-over region in between. Chain lengths are up to N=80 000, slab widths up to D=120. In order to test the analysis program and to check for finite size corrections, we actually studied three different models: (a) ordinary random walks (mimicking Theta polymers); (b) self-avoiding walks; and (c) Domb-Joyce walks with the self-repulsion tuned to the point where finite size corrections for free (unrestricted) chains are minimal. For the simulations we employ the pruned-enriched-Rosenbluth method with Markovian anticipation. In addition to the partition sum (which gives us a direct estimate of the forces exerted onto the walls), we measure the density profiles of monomers and of end points transverse to the slab, and the radial extent of the chain parallel to the walls. All scaling laws and some of the universal amplitude ratios are compared to theoretical predictions.
Critical points for the percolation process, for statistical-mechanical models of ferromagnets, and for self-avoiding and self-interacting walks are briefly discussed. The construction of expansions for such critical points in powers of 1/d, where d is the dimensionality of the underlying hypercubic lattices, is reviewed. Corresponding expansions for the transition points, T c (d), of Ising model spin glasses with arbitrary symmetric distributions of couplings are derived to order 1/d ; for the model results correct to fifth order are obtained. Numerical results are presented for d = 3, 4, . . . , 8; the lower critical dimensionality appears to be about d< = 2.5.
In [BEI92] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals . If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) βc
, the Green's function behaves like the free one. Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times , which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice ℤ4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green's function, and requires detailed properties of the Green's function throughout a sector of the complex β plane. These estimates are derived in a companion paper [BI02].
Monte Carlo methods often are and always should be part of the normal stock‐in‐trade of the ordinary practising statistician. This paper is an introduction to such methods, with an emphasis on those that require little or no calculating machinery. Three worked numerical examples, arranged in order of increasing difficulty, illustrate some of the more important general precepts. The last of these examples touches on some problems of self‐avoiding walks, i.e., stochastic processes without multiple points.
This article reviews the current status of upper and lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice ℤ d for dimensions d≥2.
The high-temperature expansions of the partition function Z and susceptibility χ of the Ising model and the number of self-avoiding walks cn and polygons pn are obtained exactly up to the eleventh order (in "bonds" or "steps") for the general d-dimensional simple hypercubical lattices. Exact expansions of lnZ and χ in power of 1/q where q=2d, and 1/σ where σ=2d-1, for T>T0 are derived up to the fifth order. The zero-order terms are the Bragg-Williams and Bethe approximations, respectively. The Ising critical point is found to have the expansion θc=kTc/2dJ=1-q-1-11/3q-2-41/3q-3-2134/45q-4-13314/15q-5-⋯, while for self-avoiding walks μ=limit of |cn|1/n as n→∞=σ[1-σ-2-2σ-3-11σ-4-62σ-5-⋯].Numerical extrapolation yields accurate estimates for θc and μ when d=2 to 6 and indicates that χ diverges as (T-Tc)-[1+δ(d)] where 3/δ(d)≃4, 12, 32±1, 80±2, 188±12, ⋯(d=2, 3⋯), and that cn≈Anαμn (n→∞) with 1/α(d)≃3, 6, 14±0.3, 32±1.5, 72±7, ⋯.
The relationship between the excluded-volume problem for a discrete random walk on a lattice and the corresponding Ising model of ferromagnetism is investigated. Systematic methods are presented for the construction of rigorous lower bounds to the limit mu=limn-->∞(cn+1cn), where cn is the number of n-step self-avoiding walks on a given lattice. In this way Temperley's conjecture that mu=(JkTC), where TC is the Curie temperature of the corresponding Ising-model ferromagnet, is disproved. The series cn for various two- and three-dimensional lattices have been enumerated exactly for values of n from ten to twenty. Extrapolation of these series, by procedures known to be valid from exact Ising-model results, yields more accurate values of mu than Wall's statistical calculations and also shows that cn~nalphamun where alpha~=13 for plane lattices and alpha~=17 for three-dimensional lattices. This means that the entropy of the nth "link" of a polymer molecule in solution should vary as deltaSn=klnmu+kalphan. The relevance of these results to the interpretation of the boundary tension of the Ising model, to the critical behavior of gases, and to the mean square size of a polymer molecule is discussed briefly.
Exact enumeration data in dimensions d=2-6 is used to evaluate the exact d-dimensional mean-square end-to-end distance R2n of a short (n
We present large statistics simulations of 3-dimensional star polymers with up to f = 80 arms and with up to 4000 monomers per arm for small values of f. They were done for the Domb-Joyce model on the simple cubic lattice. This is a model with soft core exclusion which allows multiple occupancy of sites but punishes each same-site pair of monomers with a Boltzmann factor v < 1. We use this to allow all arms to be attached at the central site, and we use the "magic" value v = 0.6 to minimize corrections to scaling. The simulations are made with a very efficient chain growth algorithm with resampling, PERM, modified to allow simultaneous growth of all arms. This allows us to measure not only the swelling (as observed from the center-to-end distances) but also the partition sum. The latter gives very precise estimates of the critical exponents gamma(f). For completeness we made also extensive simulations of linear (unbranched) polymers which give the best estimates for the exponent gamma.
The author shows how to expand critical amplitudes in inverse powers of the dimensionality d. The technique, which is rather general, is applied to the classical n-vector model, with particular emphasis on the self-avoiding walk (n=0) and Ising (n=1) cases. The existence of these expansions and of similar ones for critical points and critical constants suggests that 1/d expansions for non-universal quantities are analogous to the epsilon expansions for universal quantities.
We have computed the number, the mean-square end-to-end distance, the mean-square radius of the gyration, and the mean-square distance of a monomer from the origin, for n-step self-avoiding walks on hypercubic lattices in 4, 5, and 6 dimensions up to 19, 15, and 14 steps, respectively. Two amplitude ratios are estimated; the results for self-avoiding walks in 5 and 6 dimensions agree with the exact values for random walks in arbitrary dimensions.
We discuss the approximation of functions by the solution of certain differential equations derived from their power-series coefficients. We call these approximations integral-curve approximants, or more simply integral approximants, and find that they include as special cases many of the currently used methods. We investigate their invariance and singularity properties, and test the power of the first-order ones on a series of known functions. These integral approximants do as well, and often better, than any of the now current (nonexact) methods of approximation. We have applied them to the low-temperature Ising-model susceptibility and find, for the first time by series methods, reasonably good evidence that the low-temperature critical index is equal to the high-temperature one, as expected from the scaling and renormalization-group approaches.
Let χn(d) [resp. γ2n−1(d)] be the number of self‐avoiding walks (polygons) of n (2n − 1) steps on the integral points in d dimensions. It is known that βd = limn→∞ [χn(d)]1/n = limn→∞ [γ2n−1(d)]1/2n−1 In this paper βd is compared with βd,2r = limn→∞ [χn,2r(d)]1/n where χn,2r(d) is the number of n‐step walks on the integral points in d dimensions with no loops of 2r steps or less. In other words the walks counted in χn,2r(d) may visit the same point more than once as long as there are more than 2r steps between consecutive visits. It turns out that βd,2r−βd = O(d−r)(d→∞) and it follows in particular that 1βd = 2d−1−1/2d+O(1/d2)(d→∞) It is also shown that, for suitable constants α6 = α6(d) and α7 = α7(d),
This paper is a continuation of the companion paper [14], in which it
was proved that the standard model of self-avoiding walk in five or more
dimensions has the same critical behaviour as the simple random walk,
assuming convergence of the lace expansion. In this paper we prove the
convergence of the lace expansion, an upper and lower infrared bound,
and a number of other estimates that were used in the companion paper.
The proof requires a good upper bound on the critical point (or
equivalently a lower bound on the connective constant). In an appendix,
new upper bounds on the critical point in dimensions higher than two are
obtained, using elementary methods which are independent of the lace
expansion. The proof of convergence of the lace expansion is computer
assisted. Numerical aspects of the proof, including methods for the
numerical evaluation of simple random walk quantities such as the
two-point function (or lattice Green function), are treated in an
appendix.
The critical temperature Tc(n,d) of a classical n-component
spin model on a d-dimensional hypercubic lattice is expanded in powers
of 1d to fifth order. In the spherical-model limit, n-->∞, the
expansion is derived exactly to all orders and shown rigorously to be
only asymptotic, although Tc(∞,d) is analytic in d for
2<d<∞.
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By enumerating irreducible bridges exactly up to 40 steps, and by obtaining a lower bound to the number of bridges with less than 125 steps, a lower bound for the connective constant for square lattice self-avoiding walks of 2.62 is obtained.
For pt.I see ibid., vol.22, p.1839, (1989) For the self-avoiding walk problem, the coefficients of the chain generating function and of the generating function for the sum of square end-to-end distances have been extended to 20 terms for the triangular lattice, to 27 terms for the diamond lattice, to 21 terms for the simple cubic lattice and to 16 terms for the BCC lattice. Precise estimates of the critical points are obtained, and for the exponents we find that gamma =1.161+or-0.002 and nu =0.592+or-0.003 encompasses all the three-dimensional lattice data.
A perturbation calculation is given which implies that the susceptibility of the five and six-dimensional n-vector models can be written x approximately At-1(1+Bt12/) and x approximately At-1(1+Bt In t) respectively, independent of n. For n=0 and 1 it is shown that series analysis techniques can extract the correction-to-scaling' exponents, and that estimates of the critical temperatures and critical amplitudes can also be obtained. The correction-to-scaling exponents found are in agreement with those known to exist in the case of the spherical model.
For the self-avoiding walk problem the series expansions of the chain generating function and mean square end-to-end distance generating function have been extended to 27 steps for the square lattice and to 20 steps for the simple cubic lattice. The author develops an analysis protocol based on the method of integral approximants. Using this protocol the author finds excellent agreement with Nienhuis' exact exponent values (1984) in two dimensions gamma =1.34375 and nu =0.750. In three dimensions the author finds gamma =1.162+or-0.002 and nu =0.592+or-0.002. Accurate estimates of the critical point (reciprocal of the connective constant) for several two- and three dimensional lattices are also obtained.
The authors have extended the enumeration of square lattice and triangular lattice self-avoiding walks and their end-to-end distance by two terms. They have also calculated 17 terms in the series for the resistance of SAWs with bridges on the square lattice. Analysis of the new data allows the connective constants to be more accurately estimated, and affirms the values gamma =43/32 and v=3/4, and 1+ gamma for the exponent of the resistance series.
The authors are concerned with the number of figure eights, dumbbells and theta graphs weakly embeddable in a two-dimensional lattice. They show rigorously that, for the square lattice, the dominant limiting behaviour of the numbers of dumbbells and theta graphs is the same as the limiting behaviour of the number of self-avoiding walks and that the total number of figure eights is less than or equal to the number of rooted polygons. Numerical data are presented to suggest that this bound may be the best possible. Estimates of critical exponents for dumbbells and theta graphs are also obtained.
The method of concatenation (the addition of precomputed shorter chains to the ends of a centrally generated longer chain) has permitted the extension of the exact series for CN-the number of distinct configurations for self-avoiding walks of length N. The authors report on the leading exponent y and xc (the reciprocal of the connectivity constant) for the 2D honeycomb lattice (42 terms) 1.3437, 0.541 1968; the 2D square lattice (30 terms) 1.3436, 0.379 0520; the 3D simple cubic lattice (23 terms) 1.161 932, 0.213 4987; the 4D hypercubic (18 terms) y approximately=1, 0.147 60 and the 5D hypercubic lattice (13 terms) y<or=1.025, 0.113 05. In addition they have also evaluated the leading correction terms: honeycomb Delta approximately=1, square Delta approximately=0.85, simple cubic Delta approximately=1.0 and the 4D hypercubic logarithmic correction with delta approximately=0.25.
We have substantially extended the series for the number of self-avoiding walks and the mean-square end-to-end distance on the simple cubic lattice. Our analysis of the series gives refined estimates for the critical point and critical exponents. Our estimates of the exponents γ and ν are in good agreement with recent high-precision Monte Carlo estimates, and also with recent renormalization group estimates. Critical amplitude estimates are also given. A new, improved rigorous upper bound for the connective constant µ<4.7114 is obtained.
The numbers and mean square lengths of short, neighbour-avoiding walks on the tetrahedral and body-centred cubic lattices have been determined exactly. Using standard extrapolation techniques, estimates have been made of the connective constants and mean square length exponents for these walks. The estimate of the mean square length exponent is 1.22, but a value of 1.20 also appears plausible.
A method is developed for the analysis of critical point singularities of the form f(t) approximately A mod t mod q mod ln mod t mod p with q known. The d=4, n=0 susceptibility series is extended by two further terms, and is analysed under the assumption of a singularity of the above type with q=-1. It is found that p=0.23+or-0.04, in agreement with the calculation (p=1/4) of Larkin and Khmel'nitskii (1969). The connective constant for the model is found to be 6.7720+or-0.0005.
We consider the properties of a self-avoiding polymer chain with nearestneighbor contact energy on ad-dimensional hypercubic lattice. General theoretical arguments enable us to prescribe the exact analytic form of then-segment chain partition functionC
n
,and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind=2–6. This exact form provides the main ingredient to produce a large-n expansion ind
–1of the chain free energy through fifth order with the full dependence on the contact energy retained. The -dependent chain connectivity constant and free energy amplitude are evaluated within thed
–1expansion toO(d
–5). Our general formulation includes for the first time self-avoiding walks, neighboravoiding walks, theta, and collapsed chains as particular limiting cases.
Using an expansion based on the renormalization group philosophy we prove that for aT step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT.
The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one effectively independent sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are=0.74960.0007 ind=2 and= 0.5920.003 ind=3 (95% confidence limits), based on SAWs of lengths 200N10000 and 200N 3000, respectively.
The lace expansion has been used successfully to study the critical behaviour in high dimensions of self-avoiding walks, lattice trees and lattice animals, and percolation. In each case, the lace expansion has been an expansion along a time interval. In this paper, we introduce the lace expansion on a tree, in which ‘time’ is generalised from an interval to a tree. We develop the expansion in the context of networks of mutually-avoiding self-avoiding walks joined together with the topology of a tree, in dimensions d>4, and prove Gaussian behaviour for sufficiently spread-out networks consisting of long self-avoiding walks.
We investigate analytically the asymptotic logarithmic behavior in four dimensions (d = 4) of very long continuous polymer chains with excluded volume, described by the two-parameter Edwards model. Using renormalization theory, we calculate directly all the connected partition functions of polymer chains in this model, and obtain their logarithmic expression, including the subleading order. To first order, they are found in the universal form , and where S is the Brownian area of the chains, given in terms of their Brownian radius °R, by °R2 = dS. The partition functions 1, 2, 3, 4, and the averageend-to-end square distance R2, and square radius of gyration RG2 are calculated up to logarithmic sub-subleading order. The virial expansion in 4d of the osmotic pressure is obtained. For semi-dilute solutions, the logarithmic correction to Edwards' tree approximation for the osmotic pressure is calculated, and we find the new result Πβ ∝ C2/√lnC, where C is the monomer concentration. We give in 4d a set of geometrical quantities, characterizing a single chain: the form factor, the average 2nth power R[2n] of the end-to-end distance, the average 2nth radius of gyration RG[2n], and their universal ratio. We also consider in 4d the probability distribution for the internal distances inside a chain.
New upper bounds for the connective constant of self-avoiding walks in a hypercubic lattice are obtained by automatic generation of nite automata for counting walks with nite memory. The upper bound in dimension two is 2.679192495.
We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g)- cages, the minimum order 3-regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)-cages, and we were able to confirm that (3; 11)-cages have order 112 for the first time ever. The lower bound for a (3; 13)-cage is improved from 196 to 202 using the same approach. Also, we determined that a (3; 14)-cage has order at least 258. 1 Cages In this paper, we consider finite undirected graphs. Any undefined notation follows Bondy and Murty [7]. The girth of a graph is the size of a smallest cycle. A (r; g)- cage is an r-regular graph of minimum order with girth g. It is known that (r; g)-cages always exist [11]. Some nice pictures of small cages are given in [9, pp. 54-58]. The classification of the cages has attracted much interest amongst the graph theory community, and many of these have special nam...