  A particle subject to successive, random displacements is said to execute a
random walk (in position or some other coordinate). The mathematical properties
of random walks have been very thoroughly investigated, and the model is used
in many areas of science and engineering as well as other fields such as
finance and the life sciences. This letter describes a phenomenon occurring in
a natural extension of this model: we consider the motion of a large number of
particles subject to successive random displacements which are correlated in
space, but not in time. If these random displacements are smaller than their
correlation length, the trajectories coalesce onto a decreasing number of
trails. This surprising effect is explained and quantitative results are
obtained. Various possible realisations are discussed, ranging from coalescence
of the tracks of water droplets blown off a windshield to migration patterns of
animals.
