  Consider a medium characterized by N points whose coordinates are randomly
generated by a uniform distribution along the edges of a unitary d-dimensional
hypercube. A walker leaves from each point of this disordered medium and moves
according to the deterministic rule to go to the nearest point which has not
been visited in the preceding \mu steps (deterministic tourist walk). Each
trajectory generated by this dynamics has an initial non-periodic part of t
steps (transient) and a final periodic part of p steps (attractor). The
neighborhood rank probabilities are parameterized by the normalized incomplete
beta function I_d = I_{1/4}[1/2,(d+1)/2]. The joint distribution
S_{\mu,d}^{(N)}(t,p) is relevant, and the marginal distributions previously
studied are particular cases. We show that, for the memory-less deterministic
tourist walk in the euclidean space, this distribution is:
S_{1,d}^{(\infty)}(t,p) = [\Gamma(1+I_d^{-1})
(t+I_d^{-1})/\Gamma(t+p+I_d^{-1})] \delta_{p,2}, where t=0,1,2,...,\infty,
\Gamma(z) is the gamma function and \delta_{i,j} is the Kronecker's delta. The
mean field models are random link model, which corresponds to d \to \infty, and
random map model which, even for \mu = 0, presents non-trivial cycle
distribution [S_{0,rm}^{(N)}(p) \propto p^{-1}]: S_{0,rm}^{(N)}(t,p) =
\Gamma(N)/\{\Gamma[N+1-(t+p)]N^{t+p}\}. The fundamental quantities are the
number of explored points n_e=t+p and I_d. Although the obtained distributions
are simple, they do not follow straightforwardly and they have been validated
by numerical experiments.
