Self-avoiding Walks Research Articles

On self-avoiding polygons and walks: The snake method via pattern fluctuation

For d ≥ 2 d \geq 2 and n ∈ N n \in \mathbb {N} , let W n \mathsf {W}_n denote the uniform law on self-avoiding walks of length n n beginning at the origin in the nearest-neighbour integer lattice Z d \mathbb {Z}^d , and write Γ \Gamma for a W n \mathsf {W}_n -distributed walk. We show that in the closing probability W n ( | | Γ n | | = 1 ) \mathsf {W}_n \big ( \vert \vert \Gamma _n \vert \vert = 1 \big ) that Γ \Gamma ’s endpoint neighbours the origin and is at most n − 1 / 2 + o ( 1 ) n^{-1/2 + o(1)} in any dimension d ≥ 2 d \geq 2 . The method of proof is a reworking of that in [Ann. Probab. 44 (2016), pp. 955–983], which found a closing probability upper bound of n − 1 / 4 + o ( 1 ) n^{-1/4 + o(1)} . A key element of the proof is made explicit and called the snake method. It is applied to prove the n − 1 / 2 + o ( 1 ) n^{-1/2 + o(1)} upper bound by means of a technique of Gaussian pattern fluctuation.

A lattice model for the interpretation of oligonucleotide hybridization experiments.

We present a lattice model developed to interpret oligonucleotide hybridization experiments beyond the two-state, all-or-none description. Our model is a statistical extension of the nearest-neighbor model in which all possible combinations of broken and intact base pairs in the duplex state are considered explicitly. The conformational degrees of freedom of unpaired nucleotides in the single-strand or duplex state are modeled as self-avoiding walks of the polymer chain on a cubic lattice. Translational entropy and concentration effects are modeled through a coarser lattice of single-strand sized sites. Introducing a single free parameter for the excess entropy per unpaired nucleotide results in reasonable agreement with experiment. While the model provides a generally applicable tool, we illustrate specifically how it is used to interpret equilibrium and nonequilibrium infrared spectroscopy measurements and validate that the model correctly captures sequence and length dependent effects for sequences up to 18 nucleotides. Model predictions are directly related to experiments through computed melting curves. Calculated free energy surfaces offer insight into the interpretation of temperature-jump measurements of oligonucleotide dehybridization. The model captures the interplay between configurational variation and the enthalpic stabilization of base pairing contacts in the context of a minimalist statistical description of DNA hybridization and offers useful insight beyond the simplest all-or-none picture.

Lower bounds on the localisation length of balanced random quantum walks

We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2d). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as d^2. On the cubic lattice, the method yields the lower bound 1/ln(2) for all d, and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2d)

A shape theorem for the scaling limit of the IPDSAW at criticality

In this paper we give a complete characterization of the scaling limit of the critical Interacting Partially Directed Self-Avoiding Walk (IPDSAW) introduced in Zwanzig and Lauritzen [J. Chem. Phys. 48 (1968) 3351]. As the system size $L\in\mathbb{N}$ diverges, we prove that the set of occupied sites, rescaled horizontally by $L^{2/3}$ and vertically by $L^{1/3}$ converges in law for the Hausdorff distance toward a nontrivial random set. This limiting set is built with a Brownian motion $B$ conditioned to come back at the origin at $a_{1}$ the time at which its geometric area reaches $1$. The modulus of $B$ up to $a_{1}$ gives the height of the limiting set, while its center of mass process is an independent Brownian motion. Obtaining the shape theorem requires to derive a functional central limit theorem for the excursion of a random walk with Laplace symmetric increments conditioned on sweeping a prescribed geometric area. This result is proven in a companion paper Carmona and Petrelis (2017).

Knotting statistics for polygons in lattice tubes**Dedicated to Stuart Whittington on the occasion of his 75th birthday.

We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution.Using these methods we accurately estimate the growth rates of unknotted polygons in the and tubes, and confirm that these are the same for any fixed knot-type K. We also confirm that the entropic exponent for unknots is the same as that of all polygons, and that the exponent for fixed knot-type K depends only on the number of prime factors in the knot decomposition of K. For the simplest knot-types, this leads to a good approximation for the polygon size at which the probability of the given knot-type is maximized, and in some cases we are able to sample sufficiently long polygons to observe this numerically.

Self-avoiding pruning random walk on signed network

A signed network represents how a set of nodes are connected by two logically contradictory types of links: positive and negative links. In a signed products network, two products can be complementary (purchased together) or substitutable (purchased instead of each other). Such contradictory types of links may play dramatically different roles in the spreading process of information, opinion, behaviour etc. In this work, we propose a self-avoiding pruning (SAP) random walk on a signed network to model e.g. a user’s purchase activity on a signed products network. A SAP walk starts at a random node. At each step, the walker moves to a positive neighbour that is randomly selected, the previously visited node is removed and each of its negative neighbours are removed independently with a pruning probability r. We explored both analytically and numerically how signed network topological features influence the key performance of a SAP walk: the evolution of the pruned network resulted from the node removals, the length of a SAP walk and the visiting probability of each node. These findings in signed network models are further verified in two real-world signed networks. Our findings may inspire the design of recommender systems regarding how recommendations and competitions may influence consumers’ purchases and products’ popularity.

Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for $${d\ge d_{\rm c}}$$

Consider the long-range models on $\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-\alpha}$ for some $\alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $\alpha\ne2$ that, assuming a bound on the "derivative" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{\alpha\wedge2-d}$ above the upper-critical dimension $d_c\equiv(\alpha\wedge2)m$, where $m=2$ for self-avoiding walk and the Ising model and $m=3$ for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginal case $\alpha=2$ decays as $|x|^{2-d}/\log|x|$ whenever $d\ge d_c$ (with a large spread-out parameter $L$). This solves the conjecture in the previous work, extended all the way down to $d=d_c$, and confirms a part of predictions in physics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.

Polymer collapse transition: a view from the complex fugacity plane

Distributions of zeros of the grand canonical and canonical partition functions in the complex fugacity and in the complex interaction strength plane are examined numerically for a model of rooted self-interacting self-avoiding walks on a hierarchical graph. It is shown that the pattern of zeros of the grand canonical partition function in the complex fugacity plane has a circular-like form, with the exception of zeros lying in the vicinity of the critical point. Exact values of polymer size critical exponents, in the swollen and dense phase, as well as at the point of collapse transition, are obtained by studying asymptotic behavior of zeros lying closest to the positive real axis in the fugacity plane. Distribution of zeros of the canonical partition function in the complex interaction strength plane is found to be more complicated, and an accurate estimate of crossover exponent is obtained from study of asymptotic behavior of zeros lying closest to the positive real axis in this plane. It is also shown that the next-to-leading term in the asymptotic law for canonical partition function in the dense phase has a stretched-exponential rather than the power-law form.

Calculation of \u03c0 and Classification of Self-avoiding Lattices via DNA Configuration

Numerical simulation (e.g. Monte Carlo simulation) is an efficient computational algorithm establishing an integral part in science to understand complex physical and biological phenomena related with stochastic problems. Aside from the typical numerical simulation applications, studies calculating numerical constants in mathematics, and estimation of growth behavior via a non-conventional self-assembly in connection with DNA nanotechnology, open a novel perspective to DNA related to computational physics. Here, a method to calculate the numerical value of π, and way to evaluate possible paths of self-avoiding walk with the aid of Monte Carlo simulation, are addressed. Additionally, experimentally obtained variation of the π as functions of DNA concentration and the total number of trials, and the behaviour of self-avoiding random DNA lattice growth evaluated through number of growth steps, are discussed. From observing experimental calculations of π (πexp) obtained by double crossover DNA lattices and DNA rings, fluctuation of πexp tends to decrease as either DNA concentration or the number of trials increases. Based upon experimental data of self-avoiding random lattices grown by the three-point star DNA motifs, various lattice configurations are examined and analyzed. This new kind of study inculcates a novel perspective for DNA nanostructures related to computational physics and provides clues to solve analytically intractable problems.

Force-induced desorption of uniform branched polymers

We analyze the phase diagrams of self-avoiding walk models of uniform branched polymers adsorbed at a surface and subject to an externally applied vertical pulling force which, at critical values, desorbs the polymer. In particular, models of adsorbed branched polymers with homeomorphism types, stars, tadpoles, dumbbells and combs are examined. These models generalize the earlier results on linear, ring and 3-star polymers. In the case of star polymers, we confirm a phase diagram with four phases (a free, adsorbed, ballistic, and mixed phase) first seen in Janse van Rensburg and Whittington (2018 J. Phys. A: Math. Theor. 51 204001) for 3-star polymers. The phase diagram of tadpoles may include four phases (including a mixed phase) if the tadpole is pulled from the adsorbing surface by the end vertex of its tail. If it is instead pulled from the middle vertex of its head, then there are only three phases (the mixed phase is absent). For a dumbbell pulled from the middle vertex of a ring, there are only three phases. For combs with t teeth there are four phases, independent of the value of t for all .

Use of the Complex Zeros of the Partition Function to Investigate the Critical Behavior of the Generalized Interacting Self-Avoiding Trail Model.

The complex zeros of the canonical (fixed walk-length) partition function are calculated for both the self-avoiding trails model and the vertex-interacting self-avoiding walk model, both in bulk and in the presence of an attractive surface. The finite-size behavior of the zeros is used to estimate the location of phase transitions: the collapse transition in the bulk and the adsorption transition in the presence of a surface. The bulk and surface cross-over exponents, and , are estimated from the scaling behavior of the leading partition function zeros.

Persistence length convergence and universality for the self-avoiding random walk

In this study, we show the convergence and new properties of persistence length, , for the self-avoiding random walk model (SAW) using Monte Carlo data. We generate high precision estimates of several conformational quantities with a pivot algorithm for the square, hexagonal, triangular, cubic and diamond lattices with path lengths of 103 steps. For each lattice, we accurately estimate the asymptotic limit , which corroborates the convergence of to a constant value, and allows us to check the universality on the curves. Based on the estimates we make an ansatz for dependency with lattice cell and spatial dimension, we also find a new geometric interpretation for the persistence length.

Self-attracting self-avoiding walk

This article is concerned with self-avoiding walks (SAW) on $$\mathbb {Z}^{d}$$ that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions (Ueltschi in Probab Theory Relat Fields 124(2):189–203, 2002). This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in $$d\ge 5$$ , carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander (Random Polymers, vol. 1974. Springer-Verlag, Berlin, 2009).

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

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

Curling evolution of suspended threads replicates 2D self-avoiding walk phenomena and 1D crystallization process

The conformation evolution of threads that fall freely after being released from varying altitudes was investigated and compared to behaviors generated by other well-known fundamental physical processes. It was observed that the thread conformation replicated the conformation of long polymer chains, motivating the authors to apply a 2D self-avoiding walk (SAW) model to explain their stable conformation. Strong evidence was identified that the thread conformation strongly resembles 2D SAW behavior and has scaling power comparable to that of a 2D SAW. Also, by fitting how thread end-to-end distance evolves with time, an equation was obtained that is exactly identical to the modified Avrami equation, which is usually used for explaining phase transformation processes in 1D space (D = 1). The exponential power of n = D + 1 = 2 was simply obtained in our fitting. In conclusion, it can strongly be stated that the evolution of thread conformation over time replicates the crystallization process in 1D space and an SAW in a 2D space, showing that these microscopic processes can be replicated at macroscopic scale.

Self-avoiding walk, spin systems and renormalization.

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.

On self-avoiding polygons and walks: the snake method via polygon joining

For $d \geq 2$ and $n \in \mathbb{N} $, let $\mathsf{W} _n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z} ^d$, and write $\Gamma $ for a $\mathsf{W} _n$-distributed walk. We show that the closing probability $\mathsf{W} _n \big (\vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma $’s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a general technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].

Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha $, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha $ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha $ in great generality.

Transition probabilities for infinite two-sided loop-erased random walks

The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the “middle part” of an infinite LERW loop going through $0$ and $\infty $. In this note we derive expressions for transition probabilities for this model in dimensions $d \ge 2$. For $d=2$ the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of $z \mapsto \sqrt{z} $.

Adsorption of neighbor-avoiding walks on the simple cubic lattice

We investigate neighbor-avoiding walks on the simple cubic lattice in the presence of an adsorbing surface. This class of lattice paths has been less studied using Monte Carlo simulations. Our investigation follows on from our previous results using self-avoiding walks and self-avoiding trails. The connection is that neighbor-avoiding walks are equivalent to the infinitely repulsive limit of self-avoiding walks with monomer-monomer interactions. Such repulsive interactions can be seen to enhance the excluded volume effect. We calculate the critical behavior of the adsorption transition for neighbor-avoiding walks, finding a critical temperature $T_{\text a}=3.274(9)$ and a crossover exponent $\phi=0.482(13)$, which is consistent with the exponent for self-avoiding walks and trails, leading to an overall combined estimate for three dimensions of $\phi_\text{3D}=0.484(7)$. While questions of universality have previously been raised regarding the value of adsorption exponents in three dimensions, our results indicate that the value of $\phi$ in the strongly repulsive regime does not differ from its non-interacting value. However, it is clearly different from the mean-field value of $1/2$ and therefore not super-universal.