We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all \emph{$\mathcal O(n)$-models} with mean-field interaction and magnetic field, below and at the critical temperature on the number $N$ of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrodinger operator.

中文翻译：

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平均场中的光谱差距 $${\mathcal {O}}(n)$$-Model

我们研究了具有平均场相互作用和磁场的所有 \emph{$\mathcal O(n)$-models} 的 Ginzburg-Landau 动力学发生器的光谱间隙的依赖性，低于和在临界温度下$N$ 个粒子。为了分析吉布斯测度，我们使用一步重整化方法和半经典方法来研究辅助薛定谔算子的特征值间距。