Quantum random walk on the integer lattice: examples and phenomena
نویسندگان
چکیده
We apply results of [BP07, BBBP08] to compute limiting probability profiles for various quantum random walks in one and two dimensions. Using analytic machinery we show some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjecutres are stated and computational techniques are discussed as well.
منابع مشابه
Dynamical Localization of Random Quantum Walks on the Lattice
The denomination Quantum Walks (QW for short) covers several variants of the definition we provide below . Informally, a QW describes the discrete time quantum dynamics of a particle with internal degree of freedom, the quantum walker, on a lattice. This dynamics consists in making the walker jump between neighboring sites of the lattice. The Hilbert space of the particle is the tensor product ...
متن کاملA PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کاملA Polymer Expansion for the Random Walk on the Permutation Group Associated to the Quantum Heisenberg Ferromagnet
For a long time one has associated to the Quantum Heisenberg Ferromagnet on a lattice, a random walk on the permutation group of the lattice vertices. We here present a polymer expansion for the solution of the heat equation coupled to the random walk. We work on a finite lattice, there is no question of convergence. We leave to future work bounding terms in the expansion necessary to extend th...
متن کاملPerturbation theory for random walk in asymmetric random environment
In this paper the author continues his investigation into the scaling limit of a partial difference equation on the d-dimensional integer lattice Zd, corresponding to a translation invariant random walk perturbed by a random vector field. In a previous paper he obtained a formula for the effective diffusion constant. It is shown here that for the nearest neighbor walk in dimension d ≥ 3 this ef...
متن کاملSpatial statistics for lattice points on the sphere I: Individual results
We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on t...
متن کامل