Self-avoiding Walks Research Articles

Thermal properties of a two-dimensional intrinsically curved semiflexible biopolymer**Project supported by the Minister of Science and Technology of China

We study the behaviors of mean end-to-end distance and specific heat of a two-dimensional intrinsically curved semiflexible biopolymer with a hard-core excluded volume interaction. We find the mean square end-to-end distance at large N, with N being the number of monomers. Both β and proportional constant are dependent on the reduced bending rigidity κ and intrinsic curvature c. The larger the c, the smaller the proportional constant, and 1.5 ≥ β ≥ 1. Up to a moderate κ = κc, or down to a moderate temperature T = Tc, β = 1.5, the same as that of a self-avoiding random walk, and the larger the intrinsic curvature, the smaller the κc. However, at a large κ or a low temperature, β is close to 1, and the conformation of the biopolymer can be more compact than that of a random walk. There is an intermediate regime with 1.5 > β > 1 and the transition from β = 1.5 to β = 1 is smooth. The specific heat of the system increases smoothly with increasing κ or there is no peak in the specific heat. Therefore, a nonvanishing intrinsic curvature seriously affects the thermal properties of a semiflexible biopolymer, but there is no phase transition in the system.

Location of the adsorption transition for lattice polymers

We consider various lattice models of polymers: lattice trees, lattice animals, and self-avoiding walks. The polymer interacts with a surface (hyperplane), receiving an energy reward of β for each site in the surface. It is known that there is an adsorption transition at a critical value of β. We present a new proof of the result of Hammersley et al (1982 J. Phys. A: Math. Gen. 15 539–71) that the transition occurs at a strictly positive value of β when the surface is impenetrable, i.e. when the polymer is restricted to a half-space. In contrast, for a penetrable surface, it is an open problem to prove that the transition occurs at . We reduce this problem to proving that the fraction of N-site polymers whose span is less than is not too small.

Self-avoiding walks adsorbed at a surface and pulled at their mid-point

We consider a self-avoiding walk on the d-dimensional hypercubic lattice, terminally attached to an impenetrable hyperplane at which it can adsorb. When a force is applied the walk can be pulled off the surface and we consider the situation where the force is applied at the middle vertex of the walk. We show that the temperature dependence of the critical force required for desorption differs from the corresponding value when the force is applied at the end-point of the walk. This is of interest in single molecule pulling experiments since it shows that the required force can depend on where the force is applied. We also briefly consider the situation when the force is applied at other interior vertices of the walk.

A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent \begin{document}$ν$\end{document} governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension \begin{document}$1/ν $\end{document} , which is strictly greater than \begin{document}$1$\end{document} .

Scaling limit of the uniform prudent walk

We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investigated with combinatorial techniques in [Bousquet-M\'elou, 2010], while another variant, the kinetic prudent walk has been analyzed in detail in [Beffara, Friedli and Velenik, 2010]. In this paper, we prove that the $2$-dimensional uniform prudent walk is ballistic and follows one of the $4$ diagonals with equal probability. We also establish a functional central limit theorem for the fluctuations of the path around the diagonal.

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

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

Irreversible Markov chain Monte Carlo algorithm for self-avoiding walk

We formulate an irreversible Markov chain Monte Carlo algorithm for the self-avoiding walk (SAW), which violates the detailed balance condition and satisfies the balance condition. Its performance improves significantly compared to that of the Berretti-Sokal algorithm, which is a variant of the Metropolis-Hastings method. The gained efficiency increases with the spatial dimension (D), from approximately $10$ times in 2D to approximately $40$ times in 5D. We simulate the SAW on a 5D hypercubic lattice with periodic boundary conditions, for a system with a linear size up to $L=128$, and confirm that as for the 5D Ising model, the finite-size scaling of the SAW is governed by renormalized exponents $\nu^*=2/d$ and $\gamma/\nu^*=d/2$. The critical point is determined, which is approximately $8$ times more precise than the best available estimate.

ОЦЕНКА ЧИСЛА Р2–РАЗБИЕНИЙ ПЛОСКОСТИ НА ПОЛИМИНО ЗАДАННОЙ ПЛОЩАДИ

We consider the problem about a number of \(p2\)--tilings of a plane by a given area polyominoes. A polyomino is a connected plane geometric figure formed by joining one or more unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a whole plane by polyominoes. The tiling is called \(p2\)--tiling, if each tile can be mapped to any other tile by the translation or the central symmetry, and this transformation maps the whole tiling to itself. \(p2\)-tilings are special case of regular plane tilings. Let \(t(n)\) be a number of \(p2\)--tilings of a plane by a \(n\)-area polyomino such that the lattices of periods of these tilings are sublattices of \(\mathbb{Z}^2\). It is proved that following inequality is true: \( C_12^n \leq t(n)\leq C_2n^4(2.68)^n\). To prove the lower bound we use the exact construction of required tilings. The proof of the upper bound is based on the Conway criterion of the existence of \(p2\)--tilings of a plane. Also, the upper bound depends on the theory of self-avoiding walks on the square lattice. Earlier similar results were obtained by authors for the number of lattice tilings of a plane by a given area polyomino (it's more simple type of a plane tilings by polyomino), and for the number of lattice tilings of the plane by centrosimmetrical polyomino.

Generalized approach to the non-backtracking lace expansion

The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the mathbb {Z}^d-lattice for percolation, lattice trees and lattice animals. The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in dge 11, for lattice trees in dge 16 and for lattice animals in dge 18.

Construction and Enumeration of Circuits Capable of Guiding a Miniature Vehicle

Abstract In contrast to traditional toy tracks, a patented system allows the creation of a large number of tracks with a minimal number of pieces, and whose loops always close properly. These circuits strongly resemble traditional self-avoiding polygons (whose explicit enumeration has not yet been resolved for an arbitrary number of squares) yet there are numerous differences, notably the fact that the geometric constraints are different than those of self-avoiding polygons. We present the methodology allowing the construction and enumeration of all of the possible tracks containing a given number of pieces. For small numbers of pieces, the exact enumeration will be treated. For greater numbers of pieces, only an estimation will be offered. In the latter case, a randomly construction of circuits is also given. We will give some routes for generalizations for similar problems.

Zeons, Orthozeons, and Graph Colorings

Nilpotent adjacency matrix methods have proven useful for counting self-avoiding walks (paths, trails, cycles, and circuits) in finite graphs. In the current work, these methods are extended for the first time to problems related to graph colorings. Nilpotent-algebraic formulations of graph coloring problems include necessary and sufficient conditions for k-colorability, enumeration (counting) of heterogeneous and homogeneous paths, trails, cycles, and circuits in colored graphs, and a matrix-based greedy coloring algorithm. Introduced here also are the orthozeons and their application to counting monochromatic self-avoiding walks in colored graphs. The algebraic formalism easily lends itself to symbolic computations, and Mathematica-computed examples are presented throughout.

On the growth constant for square-lattice self-avoiding walks

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. We discuss the three most successful methods for estimating the growth constant, including the most recently developed topological transfer-matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is

Regular Simple Queues of Protein Contact Maps.

A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with [Formula: see text] peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions.

Fractal Dimensions of Self-Avoiding Walks and Ising High-Temperature Graphs in 3D Conformal Bootstrap

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the $O(N)$ models from $N=1$ (Ising model) to $N=0$ (polymer). The unitarity bound below $N=1$ of the scaling dimension for the the $O(N)$-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as $N$ tends to zero, an emerging asymmetric minimum in the current central charge $C_J$ can be used to locate the CFT. It is pointed out that certain level degeneracies at the $O(N)$ CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the $\mathcal{N}=1$ supersymmetry, which could be relevant for locating the correspoinding fixed point in the phase diagram.

Counting self-avoiding walks on free products of graphs

The connective constantμ(G) of a graph G is the asymptotic growth rate of the number σn of self-avoiding walks of length n in G from a given vertex. We prove a formula for the connective constant for free products of quasi-transitive graphs and show that σn∼AGμ(G)n for some constant AG that depends on G. In the case of products of finite graphs μ(G) can be calculated explicitly and is shown to be an algebraic number.

Comparison of Critical Adsorption Points of Ring Polymers with Linear Polymers

The critical adsorption points (CAP) for ring and linear polymers are determined and compared using Monte Carlo simulations and liquid chromatography experiments. The CAP is defined as the coelution point of ring or linear polymers with different molecular weights (MW). Computational studies show that the temperature at the CAP, TCAP, for rings is higher than TCAP for linear polymers regardless of whether the chains are modeled as random walks or self-avoiding walks. The difference in the CAP can be attributed only to the architectural difference. Experimentally, four pairs of linear and ring polystyrenes (PS) of different MW were synthesized and purified. Care was taken to account for the difference between the end-groups in linear polymers and the linkage unit in ring polymers. Elution of these polymers using a C18 bonded silica stationary phase and a CH2Cl2/CH3CN mixed eluent were studied. The temperature at the coelution point, TCAP, and the coelution time at the CAP, tE,CAP, were determined for both ...

Scaling Laws of Bottle‐Brush Polymers in Dilute Solutions

The structural determination and manipulation of bottle‐brush polymers, a class of polymers with serially grafted side‐chains, is challenging due to the interplay of side‐chain and backbone interactions over various length scales. The present work performs a detailed analysis, using molecular dynamics simulation techniques, to unravel these interactions by probing the distinct rod to a flexible real‐chain with self‐avoiding walk (SAW) type crossover in the backbone static structure factor. This analysis elucidates the deviation from flexible chain behavior, while also providing a quantitative measure of persistence length, . Significantly, the results identify a trend in which is consistent with the debated theoretical prediction of , where Ns is the number of monomers in each side‐chain of the bottle‐brush polymer. image

High-precision estimate of the hydrodynamic radius for self-avoiding walks.

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=2^{25}≈34×10^{6} monomers, we find that the ratio takes the value R_{G}/R_{H}=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).

A Study of the Boltzmann Sequence-Structure Channel.

We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.

Scaling Laws in Spatial Network Formation.

Geometric constraints impact the formation of a broad range of spatial networks, from amino acid chains folding to proteins structures to rearranging particle aggregates. How the network of interactions dynamically self-organizes in such systems is far from fully understood. Here, we analyze a class of spatial network formation processes by introducing a mapping from geometric to graph-theoretic constraints. Combining stochastic and mean field analyses yields an algebraic scaling law for the extent (graph diameter) of the resulting networks with system size, in contrast to logarithmic scaling known for networks without constraints. Intriguingly, the exponent falls between that of self-avoiding random walks and that of space filling arrangements, consistent with experimentally observed scaling of the radius of gyration of protein tertiary structures with their chain length.