Diffusion-driven spreading phenomena: The structure of the hull of the visited territory

Abstract

We study the hull of the territory visited by N random walkers after t time steps. The walkers move on two-dimensional substrates, starting all from the same position. For the substrate, we consider (a). a square lattice and (b). a percolation cluster at criticality. On the square lattice, we (c). also allow for birth and death processes, where at every time step, alphaN walkers die and are removed from the substrate, and simultaneously the same number of walkers is added randomly at the positions of the remaining walkers, such that the total numbers of walkers is constant in time. We perform numerical simulations for the three processes and find that for all of them, the structure of the hull is self-similar and described by a fractal dimension d(H) that slowly approaches, with an increasing number of time steps, the value d(H)=4/3. For process (c), however, the time to approach the asymptotic value increases drastically with increasing fraction of N/alpha, and can be observed numerically only for sufficiently small values of N/alpha.

Lattice gases with a point source

“…For a single RW moving on a 2D substrate, the hull has fractal dimension 4 /3 (this was conjectured by Mandelbrot [38] and proved by Lawler, Schramm, and Werner [39]). There is convincing numerical evidence [34] that the same remains valid for the hull formed by a fixed number of RWs, and we believe that in our situation the fractal dimension is also 4 /3. One can ask topological questions, e.g., how the number of holes scales with time (in 2D), what is the genus of the surface of the visited domain (in 3D), etc.…”

“…Then the absorbing boundary condition (34) is obeyed due to symmetry, while the boundary condition (35) is valid thanks to (38). Solving (32) subject to (39) yields…”

“…This model has been further analysed and generalized in subsequent studies, see e.g. [33][34][35][36] and references therein. In our setting, the number of RWs grows with time, but one can still adopt the methods of [9,11,12] to investigate, e.g.…”

Effect of Individual Properties of Flexible Chainlike Walkers in a Many-Body System

“…Various systems have been modeled mainly based on the random-walk model, and examined mainly by numerical simulations. For example, the territory visited by many random walkers [1,2] and aggregation processes of numerous random walkers [3][4][5][6][7][8][9] have been investigated. Also, traffic flows have been modeled by using the "biased" random-walk models [10][11][12][13][14][15][16][17].…”