Self-avoiding Walk Problem Research Articles

Integrating continuous differential evolution with discrete local search for meander line RFID antenna design.

The automated design of meander line RFID antennas is a discrete self-avoiding walk (SAW) problem for which efficiency is to be maximized while resonant frequency is to be minimized. This work presents a novel exploration of how discrete local search may be incorporated into a continuous solver such as differential evolution (DE). A prior DE algorithm for this problem that incorporates an adaptive solution encoding and a bias favoring antennas with low resonant frequency is extended by the addition of the backbite local search operator and a variety of schemes for reintroducing modified designs into the DE population. The algorithm is extremely competitive with an existing ACO approach and the technique is transferable to other SAW problems and other continuous solvers. The findings indicate that careful reintegration of discrete local search results into the continuous population is necessary for effective performance.

The [formula omitted] loop model on a three-dimensional lattice

We study a class of loop models, parameterized by a continuously varying loop fugacity n, on the hydrogen peroxide lattice, which is a three-dimensional cubic lattice of coordination number 3. For integer n>0, these loop models provide graphical representations for n-vector models on the same lattice, while for n=0 they reduce to the self-avoiding walk problem. We use worm algorithms to perform Monte Carlo studies of the loop model for n=0, 0.5, 1, 1.5, 2, 3, 4, 5 and 10 and obtain the critical points and a number of critical exponents, including the thermal exponent yt, magnetic exponent yh, and loop exponent yl. For integer n, the estimated values of yt and yh are found to agree with existing estimates for the three-dimensional O(n) universality class. The efficiency of the worm algorithms is reflected by the small value of the dynamic exponent z, determined from our analysis of the integrated autocorrelation times.

Mean-field behavior of the negative-weight percolation model on random regular graphs

We investigate both analytically and numerically the ensemble of minimum-weight loops in the negative-weight percolation model on random graphs with fixed connectivity and bimodal weight distribution. This allows us to study the mean-field behavior of this model. The analytical study is based on a conjectured equivalence with the problem of self-avoiding walks in a random medium. The numerical study is based on a mapping to a standard minimum-weight matching problem for which fast algorithms exist. Both approaches yield results that are in agreement on the location of the phase transition, on the value of critical exponents, and on the absence of any sizable indications of a glass phase. By these results, the previously conjectured upper critical dimension of d(u)=6 is confirmed.

Graph-based evolutionary algorithms

Evolutionary algorithms use crossover to combine information from pairs of solutions and use selection to retain the best solutions. Ideally, crossover takes distinct good features from each of the two structures involved. This process creates a conflict: progress results from crossing over structures with different features, but crossover produces new structures that are like their parents and so reduces the diversity on which it depends. As evolution continues, the algorithm searches a smaller and smaller portion of the search space. Mutation can help maintain diversity but is not a panacea for diversity loss. This paper explores evolutionary algorithms that use combinatorial graphs to limit possible crossover partners. These graphs limit the speed and manner in which information can spread giving competing solutions time to mature. This use of graphs is a computationally inexpensive method of picking a global level of tradeoff between exploration and exploitation. The results of using 26 graphs with a diverse collection of graphical properties are presented. The test problems used are: one-max, the De Jong functions, the Griewangk function in three to seven dimensions, the self-avoiding random walk problem in 9, 12, 16, 20, 25, 30, and 36 dimensions, the plus-one-recall-store (PORS) problem with n=15,16, and 17, location of length-six one-error-correcting DNA barcodes, and solving a simple differential equation semi-symbolically. The choice of combinatorial graph has a significant effect on the time-to-solution. In the cases studied, the optimal choice of graph improved solution time as much as 63-fold with typical impact being in the range of 15% to 100% variation. The graph yielding superior performance is found to be problem dependent. In general, the optimal graph diameter increases and the optimal average degree decreases with the complexity and difficulty of the fitness landscape. The use of diverse graphs as population structures for a collection of problems also permits a classification of the problems. A phylogenetic analysis of the problems using normalized time to solution on each graph groups the numerical problems as a clade together with one-max; self-avoiding walks form a clade with the semisymbolic differential equation solution; and the PORS and DNA barcode problems form a superclade with the numerical problems but are substantially distinct from them. This novel form of analysis has the potential to aid researchers choosing problems for a test suite

Some solvable, and as yet unsolvable, polygon and walk models

One partly solvable and two solvable models of polygons are discussed. Using a simple transfer matrix approach Iwan Jensen has derived very long series expansions for the perimeter generating function of both three-choice polygons and punctured staircase polygons. In both cases it is found that all the terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We report on an analysis of the properties of the differential equations. Recently Enrica Duchi has discussed the problem of so-called prudent self-avoiding walks. We discuss the polygon analogue of this problem, and argue that the generating function for prudent polygons is unlikely to be differentiably finite, though a restricted version of the problem, called prudent polygons of the second type, is likely to be differentiably finite. The exact generating function for prudent polygons of the first type is also found.

Invariance relations for random walks on simple cubic lattices

In recent work, we have built upon seminal contributions of Montroll and Weiss to develop invariance relations for random walks on k = 2 hexagonal and square-planar lattices. In this study, we extend our approach to determine invariance relations for random walks on k = 3 finite, simple cubic lattices subject to periodic boundary conditions. Use of these invariance relations allows one to calculate estimates of the overall mean walklength before trapping, and we show that the results obtained are in excellent agreement with numerically-exact calculated values. The results and analysis presented here yield insights on the problem of self-avoiding walks on cubic lattices.

Enumerations in statistical mechanics and combinatorics

Mathematicians require proofs, mathematical physicists are content to know that a result is exact, and physicists require only a numerical approximation adequate for their purposes. This caricature is becoming increasingly inaccurate as computers become capable of refining approximations to the extent that we can make exact conjectures. Once you have a conjecture that you believe is exact, providing a proof is generally easier. Furthermore, computer programs that can provide proofs, or exact conjectures, or even conjectured analytic information, are becoming more widespread and increasingly powerful. The article focuses primarily on the methods that the author has developed that provide the raw material for such conjectures, i.e., the enumerative techniques that produce a generating function's early terms. Using polyomino enumeration and the self-avoiding walk problem as examples, the author shows how to produce enough terms of the generating function to enable soundly based conjectures about that function's analytic properties.

WINDING ANGLE DISTRIBUTION OF SELF-AVOIDING WALKS IN TWO DIMENSIONS

Winding angle problem of two-dimensional self-avoiding walks (SAWs) on a square lattice is studied intensively by the scanning Monte Carlo simulation at high, theta (Θ), and low-temperatures. The winding angle distribution PN(θ) and the even moments of winding angle [Formula: see text] are calculated for lengths of SAWs up to N = 300 and compared with the analytical prediction. At the infinite temperature (good solvent regime of linear polymers), PN(θ) is well described by either a Gaussian function or a stretched exponential function which is close to Gaussian, so, it is not incompatible with an analytical prediction that it is a Gaussian function exp [-θ2/ ln N] in terms of a variable [Formula: see text] and that [Formula: see text]. However, the results for SAWs at Θ and low-temperatures (Θ and bad solvent regime of linear polymers) significantly deviate from this analytical prediction. PN(θ) is then described much better by a stretched exponential function exp [-|θ|α/ln N] and [Formula: see text] with α = 1.54 and 1.51 which is far from being a Gaussian. We provide a consistent numerical evidence that the winding angle distribution for SAWs at the finite temperatures may not be a Gaussian function but a nontrivial distribution, possibly a stretched exponential function.

The tricritical behavior of self-interacting partially directed walks

We present the thermodynamics of two variations of the interacting partially directed self-avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values. While the discrete model has been considered previously to varying degrees of success, the continuous model we now define has not. The examination of the continuous model leads to theexact derivation of several exponents. For the discrete model some of these exponents can be calculated using a continued-fraction representation. For both models the crossover exponentφ is found to be 2/3. Moreover, we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function. Additionally, the similarities noticed previously to other models, but left unexplored, are explained with the aid of necklacing arguments.

The self-avoiding walk problem

Different approaches to the self-avoiding walk problem are reviewed. The problem first arose in the statistical physics of linear polymers in connection with the evaluation of the average size of a polymer. The probability distribution density WN(R) for the vector R connecting the end-points of an N-step self-avoiding walk is the main quantity in this problem. The equation for WN(R) seems to be invariant under the scaling transformation group. This means that the renormalization group method can be used to determine the asymptotic form of WN(R) as N → ∞.

Self-avoiding walks on fractals studied by the Monte Carlo renormalization group

The authors have applied the Monte Carlo renormalization group (MCRG) method to study the problem of self-avoiding walks on the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b, 2<or=b<or= infinity , and when b to infinity both the fractal df and spectral ds dimension approach their Euclidean value 2. They have calculated the critical exponent nu , associated with the mean square end-to-end distance, up to b=80. Their MCRG results deviate at most 0.03% from the available exact results (for 2<or=b<or=9). The obtained data show clearly that nu monotonically decreases with b and crosses the Euclidean value nu =3/4 at b approximately=27, that is, before entering the fractal to Euclidean crossover region that occurs in the limit b to infinity .

Проблема случайных блужданий без самопересечений

Проблема случайных блужданий без самопересечений

Position-space renormalization group for SAWs on the L lattice

The self-avoiding walk problem on the L lattice is treated with a position-space renormalization group approach using a 'centre-rule' weight function.

Criticality of the discrete N-vector ferromagnet in planar self-dual lattices

The authors study an extended version of the discrete N-vector (or cubic) ferromagnetic model within a real space renormalisation group approach which preserves the two-spin correlation function. The N-evolution (for real values of N) of the Wheatstone-bridge hierarchical lattice phase diagram, which presents paramagnetic intermediate (nematic-like) and ferromagnetic phases, as well as of the thermal ( nu ) and crossover ( phi ) critical exponents, is presented. The self-avoiding walk problem is recovered in the N to O limit, and the so-called 'corner rule' is re-obtained in a larger context. The Ising, N- and 2N-state Potts ferromagnets are recovered as particular cases. An interchange of stability occurs at N=N* approximately=6.9 in such a way that the 2N-state Potts special point (where all three existing phases join) is multicritical if N(N*, but only critical if N)N* (consistently phi (N*)=O). For the cubic model, nu (N) presents a maximum at N=Nmax approximately=1.5. The results are exact, for all N, for the Wheatstone-bridge hierarchical lattice, and approximate, for N<or=2, for the square lattice. Last but not least, they discuss the connection between the present approach and the phenomenological renormalisation group.

Restricted walks, stability-instability transitions, and dynamic symmetries

The principal results from applying a transition matrix approach to the problem of self-avoiding walks which Domb and Hioe obtained some years ago are recalled. Some results are then presented on two other different physical problems: one on a universal critical exponent for a class of stability-instability transitions in the classical Hamiltonian systems, and the other on the existence of a characteristic set of constants of evolution when a quantum system possesses a certain type of symmetry. The similarity in some of the key concepts and methods used in these three problems, which involve studies of the distribution of the appropriate eigenvalues and the utilization of the existing symmetry, and the dissimilarity in some of the details are noted.

On the critical behaviour of self-avoiding walks. II

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.

Self-avoiding polygons — towards an exact solution?

We argue by analogy with other exactly solved models that the self-avoiding polygon problem is likely to be simpler than the self-avoiding walk problem. By greatly extending the series expansions for the square lattice polygon generating function, we obtain very precise numerical information, as well as good evidence for a particular underlying functional form. The exact solution of the (simpler) convex polygon problem is obtained, and current work on the hexagonal lattice polygon problem is described.

Transfer matrix calculation of the exponent ? for two-dimensional self-avoiding walks

We develop two independent transfer matrix methods for the determination of the exponentγ in the two-dimensional, self-avoiding walk problem. Our first method is based on the calculation of the correlation length and uses conformal invariance. Our second method is based on the direct calculation of the moments of the order parameter distribution. Our results are in good agreement with the conjectured values.

Critical exponent of a directed self-avoiding walk

It has been shown that the angular correlation range is infinite for the directed self-avoiding walk problem on the square lattice. This result implies a one-dimensional-like critical exponent, namely nu =1.

Self-avoiding polygons on the square, L and Manhattan lattices

Transfer-matrix techniques are used to extend the self-avoiding polygon generating function on the square lattice to terms in x46, corresponding to 46 step polygons. These techniques are then extended to apply to directed square lattices, such as the L and Manhattan lattice, and the self-avoiding polygon generating function to x48 is found for these lattices. Series analysis confirms that the 'specific heat' exponent alpha =1/2 for the self-avoiding walk problem, and gives the following estimates for the connective constants: mu (SQ)=2.638155+or-0.000025, mu (L)=1.5657+or-0.0019 and mu (Man.)=1.7328+or-0.0005. Some evidence for a correction to scaling exponent Delta approximately=0.84 is found from square lattice series.