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We establish the equality of the specialization $P_\lambda(x;q,0)$ of theMacdonald polynomial at $t=0$ with the graded character $X_\lambda(x;q)$ of atensor product of "single-column" Kirillov-Reshetikhin (KR) modules foruntwisted affine Lie algebras. This is achieved by constructing two uniformcombinatorial models for the crystals associated with the mentioned tensorproducts: the quantum alcove model (which is naturally associated to Macdonaldpolynomials), and the quantum Lakshmibai-Seshadri path model. We provide anexplicit affine crystal isomorphism between the two models, and realize theenergy function in both models. In particular, this gives the first proof ofthe positivity of the $t = 0$ limit of the symmetric Macdonald polynomial inthe untwisted and non-simply-laced cases, when it is expressed as a linearcombination of the irreducible characters for a finite-dimensional simple Liesubalgebra, as well as a representation-theoretic meaning of the coefficientsin this expression in terms of degree functions.

A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X