Multifractal statistics of the local order parameter at random critical points: application to wetting transitions with disorder

Free-electron gas in the Apollonian network: multifractal energy spectrum and its thermodynamic fingerprints

We study the free-electron gas in an Apollonian network within the tight-binding framework. The scale-free and small-world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities, gaps, and minibands. After an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations, we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum. The fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat. The absence of chiral symmetry of the Apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations.

Lévy flights in nonhomogeneous media: distributed-order fractional equation approach

A jumping process, defined in terms of the Lévi distributed jumping size and the Poissonian, position-dependent waiting time with the algebraic jumping rate, is discussed on the assumption that parameters of both distributions are themselves random variables which are determined from given probability distributions. The fractional equation for the distributed Lévy order parameter mu is derived and solved. The solution is of the form of a combination of the Fox functions and simple scaling is lacking. The problem of accelerated diffusion is also discussed. The case of the distributed waiting time parameter theta is similarly solved and the solution offers a possibility to manage processes which are characterized by more general forms of the jumping rate, not only algebraic. Moreover, we mention a possibility that the parameters mu and theta are mutually dependent.