About the Random Walk from ManyInjection Points

The random walk on a discrete lattice has been analysed in completely different fields such as chemistry [1, 2], ecology [3, 4], and general physics [5, 6]. The general idea has been to insert a diffusion point at the centre of a 1D, 2D, or 3D discrete lattice and to follow the evolution from an analytical or numerical [7–9] point of view. Here we will start by analysing the stationary state of the random walk in 2D (Section 2) with only one diffusion point at the centre of the lattice. Once the structural behaviour of the visitation grid, as predicted in [8, 9], has been confirmed (in particular the existence of 3 different zones), we continue by inserting the diffusion on a 2D and 3D lattice (Section 3) from many injection points. Finally in Section 4, two applications of the developed theory concerning the field of the astrophysics are reported.