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An implementation of BFACF-style algorithms on knotted polygons in the simple cubic, face centered cubic and body centered cubic lattice is used to estimate the statistics and writhe of minimal length knotted polygons in each of the lattices. Data are collected and analysed on minimal length knotted polygons, their entropy, and their lattice curvature and writhe.
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In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice. Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices.
By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds $eta_c$ for objects of those shapes. This mapping implies $zp_{c} sim 4 eta_c = 4.51235$ in 2D and $zp_{c} sim 8 eta_c = 2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c = c/(z+b)$ we find good agreement with $c = 2^d eta_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.
We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).
We analyze a random walk strategy on undirected regular networks involving power matrix functions of the type $L^{frac{alpha}{2}}$ where $L$ indicates a `simple Laplacian matrix. We refer such walks to as `Fractional Random Walks with admissible interval $0<alpha leq 2$. We deduce for the Fractional Random Walk probability generating functions (network Greens functions). From these analytical results we establish a generalization of Polyas recurrence theorem for Fractional Random Walks on $d$-dimensional infinite lattices: The Fractional Random Walk is transient for dimensions $d > alpha$ (recurrent for $dleqalpha$) of the lattice. As a consequence for $0<alpha< 1$ the Fractional Random Walk is transient for all lattice dimensions $d=1,2,..$ and in the range $1leqalpha < 2$ for dimensions $dgeq 2$. Finally, for $alpha=2$ Polyas classical recurrence theorem is recovered, namely the walk is transient only for lattice dimensions $dgeq 3$. The generalization of Polyas recurrence theorem remains valid for the class of random walks with Levy flight asymptotics for long-range steps. We also analyze for the Fractional Random Walk mean first passage probabilities, mean first passage times, and global mean first passage times (Kemeny constant). For the infinite 1D lattice (infinite ring) we obtain for the transient regime $0<alpha<1$ closed form expressions for the fractional lattice Greens function matrix containing the escape and ever passage probabilities. The ever passage probabilities fulfill Riesz potential power law decay asymptotic behavior for nodes far from the departure node. The non-locality of the Fractional Random Walk is generated by the non-diagonality of the fractional Laplacian matrix with Levy type heavy tailed inverse power law decay for the probability of long-range moves.
The study of the properties of quantum particles in a periodic potential subject to a magnetic field is an active area of research both in physics and mathematics; it has been and it is still deeply investigated. In this review we discuss how to implement and describe tunable Abelian magnetic fields in a system of ultracold atoms in optical lattices. After discussing two of the main experimental schemes for the physical realization of synthetic gauge potentials in ultracold set-ups, we study cubic lattice tight-binding models with commensurate flux. We finally examine applications of gauge potentials in one-dimensional rings.
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