A new finite-size scaling approach to random walks

We present a new finite-size scaling method for the random walks (RW) superseding a previously widely used renormalization group approach, which is shown here to be inconsistent. The method is valid in any dimension and is based on the exact solution for the two-point correlation function and on finite-size scaling. As an example, the phase diagram is derived for the random walk in two dimensions with a surface-bulk interaction where the system has either a surface or a defect line. We also discuss an initial calculation of the corresponding phase diagram for the case of a critically diluted lattice.