The scaling window of the model d-k-CSP

Consider the model d-k-CSP which consists of a set of n variables with the domain size d, and a set of t=rnln⁡d−ln⁡p constraints where each constraint binds k distinct variables and permits pdk tuples of values. We prove that the exact phase transition of the model d-k-CSP occurs at rc=1 as long as kln⁡d=αln⁡n, where α satisfies that 12+12(2k−1)+ε≤α≤1 (ε>0 is a sufficiently small constant). More importantly, we establish the finite-size scaling about this transition. Specifically, let δ>0 be a given constant, the probability of satisfiability of the model d-k-CSP is greater than 1−δ if r≤1−Θ(1n1−νln⁡d) (ν is dependent on the control parameters) and is less than δ if r≥1+Θ(1nln⁡d).