Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk

The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with $\kappa=8/3$ leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents, but probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE$_{8/3}$.Comment: TeX file using APS REVTeX 4.0. 10 pages, 5 figures (encapsulated postscript

Surface Code Threshold in the Presence of Correlated Errors

We study the fidelity of the surface code in the presence of correlated errors induced by the coupling of physical qubits to a bosonic environment. By mapping the time evolution of the system after one quantum error correction cycle onto a statistical spin model, we show that the existence of an error threshold is related to the appearance of an order-disorder phase transition in the statistical model in the thermodynamic limit. This allows us to relate the error threshold to bath parameters and to the spatial range of the correlated errors.Comment: 5 pages, 2 figure

Self-avoiding walks crossing a square

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 $\tau < \lambda.$Comment: 27 pages, 9 figures. Paper updated and reorganised following refereein

Current reversal and exclusion processes with history-dependent random walks

A class of exclusion processes in which particles perform history-dependent random walks is introduced, stimulated by dynamic phenomena in some biological and artificial systems. The particles locally interact with the underlying substrate by breaking and reforming lattice bonds. We determine the steady-state current on a ring, and find current-reversal as a function of particle density. This phenomenon is attributed to the non-local interaction between the walkers through their trails, which originates from strong correlations between the dynamics of the particles and the lattice. We rationalize our findings within an effective description in terms of quasi-particles which we call front barriers. Our analytical results are complemented by stochastic simulations.Comment: 5 pages, 6 figure

Efficiency of the Incomplete Enumeration algorithm for Monte-Carlo simulation of linear and branched polymers

We study the efficiency of the incomplete enumeration algorithm for linear and branched polymers. There is a qualitative difference in the efficiency in these two cases. The average time to generate an independent sample of $n$ sites for large $n$ varies as $n^2$ for linear polymers, but as $exp(c n^{\alpha})$ for branched (undirected and directed) polymers, where $0<\alpha<1$. On the binary tree, our numerical studies for $n$ of order $10^4$ gives $\alpha = 0.333 \pm 0.005$. We argue that $\alpha=1/3$ exactly in this case.Comment: replaced with published versio

Mathematical Models for Estimating the Risk of vCJD Transmission

We present two different simple models for vCJD transmission by blood transfusion. Both models indicate that transfusions alone are unlikely to cause more than a few infections, unless the number of primary cases increases. To improve our models, future work should pursue data collection, empirical estimation of the model parameters, and examination of the underlying assumptions of our frameworks. Further improvements could also include examining susceptibility to vCJD infection by age group and iatrogenic infections introduced through surgical instruments. Regarding the latter, it may be worthwhile to conduct experiments to quantify the transmission of prions from an infected surgical instrument after repeated sterilization procedures

Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice

We present a high-statistics Monte Carlo determination of the exponent gamma for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is \gamma \gtapprox 1.3425(3), in agreement with the universal value 43/32 on regular lattices, but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations. We find strong corrections to scaling that seem to indicate the presence of a non-analytic exponent Delta < 1. If we assume Delta = 11/16 we find gamma = 1.3436(3), where the error is purely statistical.Comment: 24 pages, LaTeX2e, 4 figure

Equilibrium size of large ring molecules

The equilibrium properties of isolated ring molecules were investigated using an off-lattice model with no excluded volume but with dynamics that preserve the topological class. Using an efficient set of long range moves, chains of more than 2000 monomers were studied. Despite the lack of any excluded volume interaction, the radius of gyration scaled like that of a self avoiding walk, as had been previously conjectured. However this scaling was only seen for chains greater than 500 monomers.Comment: 11 pages, 3 eps figures, latex, psfi

Uncovering the topology of configuration space networks

The configuration space network (CSN) of a dynamical system is an effective approach to represent the ensemble of configurations sampled during a simulation and their dynamic connectivity. To elucidate the connection between the CSN topology and the underlying free-energy landscape governing the system dynamics and thermodynamics, an analytical soluti on is provided to explain the heavy tail of the degree distribution, neighbor co nnectivity and clustering coefficient. This derivation allows to understand the universal CSN network topology observed in systems ranging from a simple quadratic well to the native state of the beta3s peptide and a 2D lattice heteropolymer. Moreover CSN are shown to fall in the general class of complex networks describe d by the fitness model.Comment: 6 figure

A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice

We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.Comment: 17 pages, 8 figures, IoP style file