The problem of a random walk on a finite triangular lattice with a single interior source point and zig-zag absorbing boundaries is solved exactly. This problem has been previously considered intractable.
Exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends. Explicit formulae are derived for the absorption probabilities at the ends and for the expectations that a random walk w
ill visit a particular lattice site before being absorbed. Results are obtained for lattice tubes of arbitrary size and each of the regular lattice types; square, triangular and honeycomb. The results include an adjustable parameter to model the effects of strain, such as surface curvature, on the surface diffusion. Results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig-zag carbon nano-tubes with open ends.
The off-diagonal Bethe ansatz method is generalized to the integrable model associated with the $sp(4)$ (or $C_2$) Lie algebra. By using the fusion technique, we obtain the complete operator product identities among the fused transfer matrices. These
relations, together with some asymptotic behaviors and values of the transfer matrices at certain points, enable us to determine the eigenvalues of the transfer matrices completely. For the periodic boundary condition case, we recover the same $T-Q$ relations obtained via conventional Bethe ansatz methods previously, while for the off-diagonal boundary condition case, the eigenvalues are given in terms of inhomogeneous $T-Q$ relations, which could not be obtained by the conventional Bethe ansatz methods. The method developed in this paper can be directly generalized to generic $sp(2n)$ (i.e., $C_n$) integrable model.
We consider the generating function of the algebraic area of lattice walks, evaluated at a root of unity, and its relation to the Hofstadter model. In particular, we obtain an expression for the generating function of the n-th moments of the Hofstadt
er Hamiltonian in terms of a complete elliptic integral, evaluated at a rational function. This in turn gives us both exact and asymptotic formulas for these moments.
The Continuous-Time Random Walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this article we will show how the random combination of two different unbiased CTRWs can give raise to a process with clear drift, if on
e of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought as a kind of stochastic resonance. The ultimate origin of this phenomenon is the same of the Parrondos paradox in game theory
In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the
process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.