On the distribution of a random variable occurring in 1D disordered systems

The authors consider the random variable: z=1+x 1 +x 1 x 2 +x 1 x 2 x 3 + . . . , where the x i are independent, identically distributed variables. They derive some asymptotic properties of the distribution of z, which are related e.g. to the low-temperature behaviour of the random field Ising chain. For a special class of distributions of the x i , exact solutions are presented. They also study the cases where the distribution function of z exhibits a power-law fall-off modulated by a 'periodic critical amplitude'.