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Register a new userNonreversible Markov chain Monte Carlo algorithm for efficient generation of Self-Avoiding Walks
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We introduce an efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint. In two dimensions, the new algorithm slightly outperforms the two-move nonreversible Berretti-Sokal algorithm introduced by H.~Hu, X.~Chen, and Y.~Deng in cite{old}, while for three-dimensional walks, it is 3--5 times faster. The new algorithm introduces nonreversible Markov chains that obey global balance and allows for three types of elementary moves on the existing self-avoiding walk: shorten, extend or alter conformation without changing the walks length.
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We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L, L)$, and are entirely contained in the square $[0, L] times [0, L]$ on the square lattice ${mathbb Z}^2$. The number of distinct walks is known to grow as $lambda^{L^2+o(L^2)}$. We estimate $lambda = 1.744550 pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < lambda < 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$. We also consider the model in which a weight or {em fugacity} $x$ is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For $x < 1/mu$ the average length of a SAW grows as $L$, while for $x > 1/mu$ it grows as $L^2$. Here $mu$ is the growth constant of unconstrained SAW in ${mathbb Z}^2$. For $x = 1/mu$ we provide numerical evidence, but no proof, that the average walk length grows as $L^{4/3}$. We also consider Hamiltonian walks under the same restriction. They are known to grow as $tau^{L^2+o(L^2)}$ on the same $L times L$ lattice. We give precise estimates for $tau$ as well as upper and lower bounds, and prove that $tau < lambda.$
Biological systems use energy to maintain non-equilibrium distributions for long times, e.g. of chemical concentrations or protein conformations. What are the fundamental limits of the power used to hold a stochastic system in a desired distribution over states? We study the setting of an uncontrolled Markov chain $Q$ altered into a controlled chain $P$ having a desired stationary distribution. Thermodynamics considerations lead to an appropriately defined Kullback-Leibler (KL) divergence rate $D(P||Q)$ as the cost of control, a setting introduced by Todorov, corresponding to a Markov decision process with mean log loss action cost. The optimal controlled chain $P^*$ minimizes the KL divergence rate $D(cdot||Q)$ subject to a stationary distribution constraint, and the minimal KL divergence rate lower bounds the power used. While this optimization problem is familiar from the large deviations literature, we offer a novel interpretation as a minimum holding cost and compute the minimizer $P^*$ more explicitly than previously available. We state a version of our results for both discrete- and continuous-time Markov chains, and find nice expressions for the important case of a reversible uncontrolled chain $Q$, for a two-state chain, and for birth-and-death processes.
An explicit expression is derived for the scattering function of a self-avoiding polymer chain in a $d$-dimensional space. The effect of strength of segment interactions on the shape of the scattering function and the radius of gyration of the chain is studied numerically. Good agreement is demonstrated between experimental data on dilute solutions of several polymers and results of numerical simulation.
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 $$mu(mathrm{square}) = 2.63815853032790, (3).$$
Flory-Huggins theory is a mean field theory for modelling the free energy of dense polymer solutions and polymer melts. In this paper we use Flory-Huggins theory as a model of a dense two dimensional self-avoiding walk confined to a square in the square lattice. The theory describes the free energy of the walk well, and we estimate the Flory interaction parameter of the walk to be $chi_{saw} = 0.32(1)$.
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