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Using an elaborate set of simulational tools and statistically optimizedmethods of data analysis we investigate the scaling behavior of the correlationlengths of three-dimensional classical O($n$) spin models. Consideringthree-dimensional slabs $S^1\times S^1\times\mathbb{R}$, the results over awide range of $n$ indicate the validity of special scaling relations involvinguniversal amplitude ratios that are analogous to results of conformal fieldtheory for two-dimensional systems. A striking mismatch of the $n\to\infty$extrapolation of these simulations against analytical calculations is tracedback to a breakdown of the identification of this limit with the sphericalmodel.Comment: 18 pages, 9 figures, REVTeX4, slightly shortened, updated critical  exponent estimate

Monte Carlo study of the scaling of universal correlation lengths in three-dimensional O(n) spin models