  We give an algorithmic proof for the existence of tight Hamilton cycles in a
random r-uniform hypergraph with edge probability p=n^{-1+eps} for every eps>0.
This partly answers a question of Dudek and Frieze [Random Structures
Algorithms], who used a second moment method to show that tight Hamilton cycles
exist even for p=omega(n)/n (r>2) where omega(n) tends to infinity arbitrary
slowly, and for p=(e+o(1))/n (r>3).
  The method we develop for proving our result applies to related problems as
well.
