| Summary: | Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G<sub>⊥</sub> (k) ≅ a k<sup>-λ<sub></sup>⊥</sub> and G<sub>||</sub> (k) ≅ b k<sup>-λ<sub>||</sub></sup> in the ordered phase at k → 0, is performed in the three-dimensional O(n) models with n=2, 4, 10. Our aim is to test some challenging theoretical predictions, according to which the exponents λ<sub>⊥</sub> and λ<sub>||</sub> are non-trivial (3/2<λ<sub>⊥</sub><2 and 0<λ<sub>||</sub><1 in three dimensions) and the ratio b M<sup>2</sup>/a<sup>2</sup> (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are λ<sub>⊥</sub>=2 and λ<sub>||</sub>=1. Our earlier MC analysis gives λ<sub>⊥</sub>=1.955 ± 0.020 and λ<sub>||</sub> about 0.9 for the O(4) model. A recent MC estimation of λ<sub>||</sub>, assuming corrections to scaling of the standard theory, yields λ<sub>||</sub> = 0.69 ± 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding λ<sub>⊥</sub> = 1.9723(90). We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by Δ λ<sub>⊥</sub> = 0.0121(52). It is consistent with the idea that 2-λ<sub>⊥</sub> decreases for large n and tends to zero at n → ∞. We have also verified and confirmed the expected universality of b M<sup>2</sup>/a<sup>2</sup> for the O(4) model, where simulations at two different temperatures (couplings) have been performed.
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