It is claimed that calculating partition function of 3 dimensional ising model is NP-complete. But I have a question, is it saying that calculating partition function is exponential with respect to the size of lattice? But generally in physics, I think people is interested in the partition function of an infinite system, not some finite system. And that exponentiation shall not preclude the chance of getting partition function infinite 3D systems, right?
Second question: From complexity theory class that I self taught, one problem is called NP-complete only if the solution could be verify in polynomial time. So is it true that one could give a partition function for a finite 3D Ising system with a certificate which can be checked in polynomial time?
Third question: If I am not wrong, I consider the ground state problem of 3D Ising problem to be NP-hard and not NP, right?