  While completely self-avoiding quantum walks have the distinct property of
leading to a trivial unidirectional transport of a quantum state, an
interesting and non-trivial dynamics can be constructed by restricting the
self-avoidance to a subspace of the complete Hilbert space. Here, we present a
comprehensive study of three two-dimensional quantum walks, which are
self-avoiding in coin space, in position space and in both, coin and position
space. We discuss the properties of these walks and show that all result in
delocalisation of the probability distribution for initial states which are
strongly localised for a walk with a standard Grover coin operation. We also
present analytical results for the evolution in the form of limit distributions
for the self-avoiding walks in coin space and in both, coin and position space.
