Dynamics of exponential linear map in functional space

We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a probability measure $\mu$, which corresponds to the weight distribution of a link of a random graph we form a positive linear operator $\Phi$ (convolution) on distribution functions and then we analyze a family of its exponents with a parameter $\lambda$ which corresponds to connectivity of a sparse random graph. We prove that for every measure $\mu$ (\emph{i.e.}, convolution $\Phi$) and every $\lambda< e$ there exists a unique globally attracting fixed point of the operator, which yields the existence and uniqueness of the limit probability distribution on the random graph. This estimate was established earlier \cite{KarpSipser} for deterministic weight distributions (Dirac measures $\mu$) and is known as $e$-cutoff phenomena, as for such distributions and $\lambda>e$ there is no fixed point attractor. We thus establish this phenomenon in a much more general sense.