  We consider the problem of $q$-colouring a $k$-uniform random hypergraph,
where $q,k \geq 3$, and determine the rigidity threshold. For edge densities
above the rigidity threshold, we show that almost all solutions have a linear
number of vertices that are linearly frozen, meaning that they cannot be
recoloured by a sequence of colourings that each change the colour of a
sublinear number of vertices. When the edge density is below the threshold, we
prove that all but a vanishing proportion of the vertices can be recoloured by
a sequence of colourings that recolour only one vertex at a time. This change
in the geometry of the solution space has been hypothesised to be the cause of
the algorithmic barrier faced by naive colouring algorithms. Our calculations
verify predictions made by statistical physicists using the non-rigorous cavity
method.
  The traditional model for problems of this type is the random colouring
model, where a random hypergraph is chosen and then a random colouring of that
hypergraph is selected. However, it is often easier to work with the planted
model, where a random colouring is selected first, and then edges are randomly
chosen which respect the colouring. As part of our analysis, we show that up to
the condensation phase transition, the random colouring model is contiguous
with respect to the planted model. This result is of independent interest.
