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

The authors consider the random variable: z=1+x1+x1x2+x1x2x3+ . . . , where the xi 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 xi, 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'.

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