The resolution complexity of random constraint satisfaction problems

We consider random instances of constraint satisfaction problems where each variable has domain size d, and each constraint contains t restrictions on k variables. For each (d, k, t) we determine whether the resolution complexity is a.s. constant, polynomial or exponential in the number of variables. For a particular range of (d, k, t) we determine a sharp threshold for resolution complexity where the resolution complexity drops from a.s. exponential to a.s. polynomial when the clause density passes a specific value.