Eternal inflation and localization on the landscape

We model essential features of eternal inflation on the landscape of a dense discretuum of vacua by the potential V (phi) = V-0 + delta V(phi), where vertical bar delta V(phi)vertical bar << V-0 is random. We find that the diffusion of the distribution function rho(phi, t) of the inflaton expectation value in different Hubble patches may be suppressed due to the effect analogous to the Anderson localization in disordered quantum systems. At late times only the localized part of the distribution function rho(phi, t) survives which leads to dynamical selection principle on the landscape. As a result, there exist such islands on the landscape that the probability to measure any but a small value of the cosmological constant in a given Hubble patch is exponentially suppressed.