Statistical properties of randomly broken objects and of multivalley structures in disordered systems

The statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common. For all these problems, if W, denotes the weight of the sth piece, we show that the probability distributions PI( W,) of the largest piece W,, P2( W,) of the second largest piece, and n( Y) of Y = 1, Wi always have singularities at W, = l/n, W2 = l/n and Y = Ijn, n = 1,2,3,. . . .