[2.3.1] Consider first the integral equation of motion for the CTRW-model [9, 10]. [2.3.2] The probability density obeys the integral equation

(3) |

where denotes the probability for a displacement in each single step, and is the waiting time distribution giving the probability density for the time interval between two consecutive steps. [2.3.3] The transition probabilities obey , and the function is the survival probability at the initial position which is related to the waiting time distribution through

(4) |

[page 3, §0] Fourier-Laplace transformation leads to the solution in Fourier-Laplace space given as [10]

(5) |

where is the Fourier-Laplace transform of and similarly for and .

[3.1.1] Two lattice models with different waiting time density will be considered. [3.1.2] In the first model the waiting time density is chosen as the one found in [1, 2]

(6) |

where is the characteristic time, and

(7) |

is the generalized Mittag-Leffler function [14]. [3.1.3] In the second model the waiting time density is chosen as

(8) |

where , and is a suitable dimensional constant.

[3.2.1] The waiting time density differs only little from as shown graphically in Figure 1.

[3.2.2] Note that both models have a long time tail of the form given in eq. (2), and the average waiting time diverges.

[3.3.1] For both models the spatial transition probabilities are chosen as those for nearest-neighbour transitions (Polya walk) on a -dimensional hypercubic lattice given as

(9) |

where is the -th unit basis vector generating the lattice, is the lattice constant and for and for .