Phase diagrams of soluble multi-spin glass models

We include p-spin interactions in a spherical version of a soluble mean-field spin-glass model proposed by van Hemmen. Due to the simplicity of the solutions, which do not require the use of the replica trick, we are able to carry out a detailed investigation of a number of special situations. For p 3, there appear first-order transitions between the paramagnetic and the ordered phases. In the presence of additional ferromagnetic interactions, we show that there is no stable mixed phase, with both ferromagnetic and spin-glass properties.