5 research outputs found
An explicit solution for the logistic map
Abstract An explicit functional integral solution for the logistic map is presented. Furthermore, the discrete nature of this equation is used to explicitly calculate the corresponding Radon-Nikodym derivatives. This enables us to represent the solution as a multidimensional integral
Critical dimensions for random walks on random-walk chains
The probability distribution of random walks on linear structures generated
by random walks in -dimensional space, , is analytically studied
for the case . It is shown to obey the scaling form
, where is
the density of the chain. Expanding in powers of , we find that
there exists an infinite hierarchy of critical dimensions, ,
each one characterized by a logarithmic correction in . Namely, for
, ; for ,
; for , ; for , ; for , , {\it etc.\/} In particular, for
, this implies that the temporal dependence of the probability density of
being close to the origin .Comment: LATeX, 10 pages, no figures submitted for publication in PR