Beyond One Dimension

This chapter is devoted to the first-passage properties of fractal and nonfractal networks including the Cayley tree, hierarchically branched trees, regular and hierarchical combs, hierarchical blob structures, and other networks. One basic motivation for extending our study of first passage to these geometries is that many physical realizations of diffusive transport, such as hopping conductivity in amorphous semiconductors, gel chromatography, and hydrodynamic dispersion, occur in spatially disordered media. For general references see, e.g., Havlin and ben-Avraham (1987), Bouchaud and Georges (1990), and ben-Avraham and Havlin (2000). Judiciously constructed idealized networks can offer simple descriptions of these complex systems and their first-passage properties are often solvable by exact renormalization of the master equations.

In the spirit of simplicity, we study first passage on hierarchical trees, combs, and blobs. The hierarchical tree is an iterative construction in which one bond is replaced with three identical bonds at each stage; this represents a minimalist branched structure. The comb and the blob structures consist of a main backbone and an array of sidebranches or blob regions where the flow rate is vanishingly small. By varying the relative geometrical importance of the sidebranches (or blobs) to the backbone, we can fundamentally alter the first-passage characteristics of these systems.

When transport along the backbone predominates, first-passage properties are essentially one dimensional in character. For hierarchical trees, the role of sidebranches and the backbone are comparable, leading to a mean first-passage time that grows more quickly than the square of the system length. As might be expected, this can be viewed as the effective spatial dimension of such structures being greater than one.