Self-avoiding walks and polymer adsorption: edge and vertex weighting

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks (SAW) interacting with (alternate) sites on the surface of the honeycomb lattice is A key identity used in that proof depends on the existence of a parafermionic observable for SAW interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also prove the corresponding result for the edge-weighted case for the honeycomb lattice.

It is shown that the mean-field renormalisation group technique of Indekeu et al. (1982) may be applied directly to the self-avoiding walk (SAW) and percolation problems (without reference to the n to 0 limit of interacting spin systems). Numerical results for the SAW and bond and site percolation (both directed and undirected) problems on the square lattice, obtained by using a variety of small cells, are reported. An isotropic transformation for the percolation problems on the d-dimensional hypercubic lattice is also discussed.

We consider the phenomena of entanglement of the two interacting self‐avoiding walks (SAW) situated in a member of the three‐dimensional Sierpinski Gasket (SG) fractal family. We focus our attention to determine number of point contacts between the two SAW paths M, which turns out to be a set of power laws whose characteristics depend predominantly on the interactions between SAW steps. The phase diagrams have been establised and corresponding values of the contact critical exponents φ, associated with the two‐path mutual contacts, have been found.

The three-state antiferromagnetic Potts model on d-dimensional hypercubic lattices is studied, where d is 2, 3, 4, 5 and 6. The Hamiltonian is expressed by the spin Hamiltonian with spin 1. The staggered susceptibility is calculated by the high temperature series expansion. By use of the ratio method and the Pade approximation, the critical temperature and the critical exponent γ are obtained. It is concluded that the three-state antiferromagnetic Potts model exhibits the second-order transition when d ≥3.

Study of universality class for site-bond correlated diluted spin systems in the n-vector model

Polymers in random media: An introduction

Multi-way graph partition by stochastic probe

We explore the influence of a hydrophilic polymer, polyethylene glycol (PEG), on the structure and the percolation dynamics of reverse microemulsions (ME) stabilized by an anionic surfactant AOT (sodium bis(2-ethylhexyl) sulfosuccinate). The percolation transition of MEs is probed using dielectric relaxation spectroscopy (DRS). Notably, an increase in percolation temperature is observed by the incorporation of PEG-polymer into larger ME droplets which is explained by considering the model of polymer adsorption at surfactant-water interface. The stability of the droplet phase of microemulsion after the incorporation of PEG is confirmed by small-angle neutron scattering (SANS) experiment. Further, a net decrease in percolation transition temperature is observed with the addition of PEG polymer for smaller ME droplets and is discussed in relation with the destabilization of droplets owing to the polymer induced bridging and the associated clustering of droplets. We conjecture that the adsorption of PEG polymer chains at the surfactant-water interface as well as the PEG-induced bridging of droplets are due to the strong ion-dipole interaction between anionic head group of AOT surfactant and dipoles present in PEG polymer chains.

We study the polymer system consisting of two polymer chains situated in a fractalcontainer that belongs to the three-dimensional Sierpinski gasket (3D SG) family offractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact,2D SG fractals. The two-polymer system is modeled by two interacting self-avoiding walks(SAW), one of them representing a 3D floating polymer, while the other corresponds to achain confined to one of the four 2D SG boundaries (with no monomer in the bulk).We assume that the studied system is immersed in a poor solvent inducing theintra-chain interactions. For the inter-chain interactions we propose two models:in the first model (ASAW) the SAW chains are mutually avoiding, whereas inthe second model (CSAW) chains can cross each other. By applying an exactrenormalization group (RG) method, we establish the relevant phase diagrams forb = 2,3 and 4 members of the 3D SG fractal family for the model with avoiding SAWs, and forb = 2 and 3 fractals for the model with crossing SAWs. Also, at the appropriate transition fixedpoints we calculate the contact critical exponents, associated with the number of contactsbetween monomers of different chains. Throughout this paper we compare results obtainedfor the two models and discuss the impact of the topology of the underlying lattices onemerging phase diagrams.

We consider optimization problems on weighted graphs where vertex and edge weights are polynomial functions of a parameter λ. We show that, if a problem satisfies certain regularity properties and the underlying graph has bounded tree-width, the number of changes in the optimum solution is polynomially bounded. We also show that the description of the sequence of optimum solutions can be constructed in polynomial time and that certain parametric search problems can be solved in O(n log n) time, where n is the number of vertices in the graph.KeywordsComposition OperatorSteiner TreeParse TreeLower EnvelopeOuterplanar GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

The occurrence of knots in a model of a ring polymer interacting with a surface is considered. The polymer is described by a self-avoiding polygon (SAP) and interacts with the surface through a short-range interaction. It is proven rigorously that, for all non-zero temperatures, all except exponentially few SAPs contain a knot. We also show that the average knot complexity grows at least linearly with the length of the polymer, for sufficiently long polymers.

We study a modified version of the stochastic susceptible-infected-refractory-susceptible (SIRS) model by employing a nonlinear (exponential) reinforcement in the contagion rate and no diffusion. We run simulations for complete and random graphs as well as d-dimensional hypercubic lattices (for d=3,2,1). For weak nonlinearity, a continuous nonequilibrium phase transition between an absorbing and an active phase is obtained, such as in the usual stochastic SIRS model [Joo and Lebowitz, Phys. Rev. E 70, 036114 (2004)]. However, for strong nonlinearity, the nonequilibrium transition between the two phases can be discontinuous for d>or=2, which is confirmed by well-characterized hysteresis cycles and bistability. Analytical mean-field results correctly predict the overall structure of the phase diagram. Furthermore, contrary to what was observed in a model of phase-coupled stochastic oscillators with a similar nonlinearity in the coupling [Wood, Phys. Rev. Lett. 96, 145701 (2006)], we did not find a transition to a stable (partially) synchronized state in our nonlinearly pulse-coupled excitable elements. For long enough refractory times and high enough nonlinearity, however, the system can exhibit collective excitability and unstable stochastic oscillations.

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We discuss generation of series expansions for Ising spin-glasses with a symmetric $\pm J$ (i.e. bimodal) distribution on d-dimensional hypercubic lattices using linked-cluster methods. Simplifications for the bimodal distribution allow us to go to higher order than for a general distribution. We discuss two types of problem, one classical and one quantum. The classical problem is that of the Ising spin glass in a longitudinal magnetic field, $h$, for which we obtain high temperature series expansions in variables $\tanh(J/T)$ and $\tanh(h/T)$. The quantum problem is a $T=0$ study of the Ising spin glass in a transverse magnetic field $h_T$ for which we obtain a perturbation theory in powers of $J/h_T$. These methods require (i) enumeration and counting of \textit{all} connected clusters that can be embedded in the lattice up to some order $n$, and (ii) an evaluation of the contribution of each cluster for the quantity being calculated, known as the weight. We discuss a general method that takes the much smaller list (and count) of all no free-end (NFE) clusters on a lattice up to some order $n$, and automatically generates all other clusters and their counts up to the same order. The weights for finite clusters in both cases have a simple graphical interpretation that allows us to proceed efficiently for a general configuration of the $\pm J$ bonds, and at the end perform suitable disorder averaging. The order of our computations is limited by the weight calculations for the high-temperature expansions of the classical model, while they are limited by graph counting for the $T=0$ quantum system. Details of the calculational methods are presented.