Abstract: For the scalar two-phase (elliptic) Allen-Cahn equation, it is well-known that the diffuse interface is closely connected to minimal surfaces. Similarly, in the vector-valued setting, the coexistence of three or more phases is linked to minimal cones arising from the minimizing partition problem. In this talk, I will discuss some recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity. Furthermore, we prove the uniqueness of the blow-down limit at infinity by deriving precise estimates for the location and size of the diffuse interface. We also show that the solution is almost invariant along the diffuse interface and, at infinity, closely resembles one dimensional heteroclinic connections between two energy wells. Our proof is primarily variational and does not assume any symmetry of the solution. These results are based on joint works with Nicholas Alikakos.