How fast does a random walk cover a torus?

We present high statistics simulation data for the average time $\langle T_{\rm cover}(L)\rangle$ that a random walk needs to cover completely a 2-dimensional torus of size $L\times L$. They confirm the mathematical prediction that $\langle T_{\rm cover}(L)\rangle \sim (L \ln L)^2$ for large $L$, but the prefactor {\it seems} to deviate significantly from the supposedly exact result $4/\pi$ derived by A. Dembo {\it et al.}, Ann. Math. {\bf 160}, 433 (2004), if the most straightforward extrapolation is used. On the other hand, we find that this scaling does hold for the time $T_{\rm N(t)=1}(L)$ at which the average number of yet unvisited sites is 1, as also predicted previously. This might suggest (wrongly) that $\langle T_{\rm cover}(L)\rangle$ and $T_{\rm N(t)=1}(L)$ scale differently, although the distribution of rescaled cover times becomes sharp in the limit $L\to\infty$. But our results can be reconciled with those of Dembo {\it et al.} by a very slow and {\it non-monotonic} convergence of $\langle T_{\rm cover}(L)\rangle/(L \ln L)^2$, as had been indeed proven by Belius {\it et al.} [Prob. Theory \& Related Fields {\bf 167}, 1 (2014)] for Brownian walks, and was conjectured by them to hold also for lattice walks.Comment: 4 pages, 9 figures; to be published in Phys. Rev.

The RANLUX generator: resonances in a random walk test

Using a recently proposed directed random walk test, we systematically investigate the popular random number generator RANLUX developed by Luescher and implemented by James. We confirm the good quality of this generator with the recommended luxury level. At a smaller luxury level (for instance equal to 1) resonances are observed in the random walk test. We also find that the lagged Fibonacci and Subtract-with-Carry recipes exhibit similar failures in the random walk test. A revised analysis of the corresponding dynamical systems leads to the observation of resonances in the eigenvalues of Jacobi matrix.Comment: 18 pages with 14 figures, Essential addings in the Abstract onl

The Length of an SLE - Monte Carlo Studies

The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that if this fractal variation is used to parametrize the SLE, then the parametrized curves have the same distribution as the curves in the scaling limit of the lattice models with their natural parametrization.Comment: 18 pages, 10 figures. Version 2 replaced the use of "nu" for the "growth exponent" by 1/d_H, where d_H is the Hausdorff dimension. Various minor errors were also correcte

Winding Angle Distributions for Random Walks and Flux Lines

We study analytically and numerically the winding of a flux line around a columnar defect. Reflecting and absorbing boundary conditions apply to marginal or repulsive defects, respectively. In both cases, the winding angle distribution decays exponentially for large angles, with a decay constant depending only on the boundary condition, but not on microscopic features. New {\it non-universal} distributions are encountered for {\it chiral} defects which preferentially twist the flux line in one direction. The resulting asymmetric distributions have decay constants that depend on the degree of chirality. In particular, strong chirality encourages entanglements and leads to broad distributions. We also examine the windings of flux lines in the presence of point impurities (random bonds). Our results suggest that pinning to impurities reduces entanglements, leading to a narrow (Gaussian) distribution.Comment: 12 pages Revtex and 9 postscript figure

Modelling diffusional transport in the interphase cell nucleus

In this paper a lattice model for diffusional transport of particles in the interphase cell nucleus is proposed. Dense networks of chromatin fibers are created by three different methods: randomly distributed, non-interconnected obstacles, a random walk chain model, and a self avoiding random walk chain model with persistence length. By comparing a discrete and a continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter particle diffusion. The influence of the 3D geometry of the fiber network on the particle diffusion is investigated in detail, while varying occupation volume, chain length, persistence length and walker size. It is shown that adjacency of the monomers, the excluded volume effect incorporated in the self avoiding random walk model, and, to a lesser extent, the persistence length, affect particle diffusion. It is demonstrated how the introduction of the effective chain occupancy, which is a convolution of the geometric chain volume with the walker size, eliminates the conformational effects of the network on the diffusion, i.e., when plotting the diffusion coefficient as a function of the effective chain volume, the data fall onto a master curve.Comment: 9 pages, 8 figure