Asymptotics of Minimizers for Ginzburg--Landau-type Functionals in High Dimensions

We investigate local minimizers of Ginzburg--Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized energy measures converge to an $(n-2)$-rectifiable measure associated with a stationary varifold, with quantized density determined by the homotopy classes of the vacuum manifold. Away from the support of the $(n-2)$-rectifiable measure, the minimizers converge strongly in $H^1_{\text{loc}}$ to a minimizing harmonic map, which is smooth outside an $(n-3)$-rectifiable singular set.