Replica Theory and the Geometry of Symmetry Breaking

In the study of Replica Symmetry Breaking (RSB), one is led to consider functions f of pseu-domatrices, i.e. of matrices of order , with a real, rather than an integer, number. We propose a mathematically rigorous deenition of pseudomatrices, and show that from this it follows that the minimum of f over the space of pseudomatrices Q(x; y) which depend only on (x ?y) is also a critical point for f; this corresponds to a property which is usually assumed without proof in the study of RSB. We also nd that generic bifurcations from such a minimum lead to minima corresponding to periodic quasimatrices. These results are obtained through use of Michel's theory of symmetry breaking.