The Intermediate Regime

Abstract

When h_ex ∼ H ⁰_c1 i.e., $\frac{{h_{ex} }} {{\left| {log \varepsilon } \right|}} \to \lambda = \frac{1} {{2\left| {\underline {\xi _0 } } \right|}}$ , then from Theorem 7.2, we get that the limiting minimizer is h₀ hence μ* = 0. Moreover, comparing the lower bounds (7.58) and (7.59) to the upper bound of Theorem 7.1, we find $\frac{{\sum\nolimits_i {\left| {d_i } \right|} }} {{h_{ex} }} \to 0$ , which means that the number of vortices is o(h_ex). In other words, for energy-minimizers, vortices first appear for $\frac{{h_{ex} }} {{\left| {log \varepsilon } \right|}} \to \frac{1} {{2\left| {\underline {\xi _0 } } \right|}}$ , or h_ex ∼ H ⁰_c1 , and next to Λ (defined in (7.5)), and the vorticity mass is much smaller than h_ex. The analysis of Chapter 7 does not give us the optimal number n of vortices nor the full asymptotic expansion of the first critical field. Thus, a more detailed study will be necessary in this regime $h_{ex} \sim \frac{{\left| {\left. {log \varepsilon } \right|} \right.}} {{2\left| {\left. {\xi _0 } \right|} \right.}}$ , in which n ≪ h_ex. We will prove that the vortices, even though their number may be diverging, all concentrate around Λ (generically a single point) but that after a suitable blow-up, they tend to arrange in a uniform density on a subdomain of ℝ², in order to minimize a limiting interaction energy I defined on probability measures.