  Asymptotic dynamics of a Hadamard walk of two non-interacting quantum
particles on a dynamically percolated finite line or a circle is investigated.
We construct a basis of the attractor space of the corresponding random-unitary
dynamics and prove the completeness of our solution. In comparison to the
one-particle case, the structure of the attractor space is much more complex,
resulting in intriguing asymptotic dynamics. General results are illustrated on
two examples. First, for circles of length not divisible by 4 the boundary
conditions reduces the number of attractors considerably, allowing for fully
analytic solution. Second, we investigate line of length 4 and determine the
asymptotic cycle of reduced coin states and position distributions, focusing on
the correlations between the two particles. Our results show that a random
unitary evolution, which is a combination of quantum dynamics and a classical
stochasticity, leads to correlations between initially uncorrelated particles.
This is not possible for purely unitary evolution of non-interacting quantum
particles. The shared dynamically percolated graph can thus be considered as a
weak form of interaction.
