Narrow Results

With the aim of describing a general benchmark for several complex systems, we analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes. The model is described by an infinite series of order parameters (the multi-overlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multi-overlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line.

We discuss the competition between the Kondo effect, the spin glass state and a magnetic order observed in disordered Cerium systems. We present firstly the experimental situation of disordered alloys such as CeNi_1-xCu_x and then the different theoretical approaches based on the Kondo lattice model, with different descriptions of the intersite exchange interaction for the spin glass. After the gaussian approach of the Sherrington-Kirkpatrick model, we discuss the Mattis and the van Hemmen models. Then, we present simple cluster calculations in order to describe the percolative evolution of the clusters from the cluster spin glass to the inhomogeneous ferromagnetic order recently observed in CeNi_1-xCu_x disordered alloys and finally we discuss the effect of random and transverse magnetic field.