Commun. Comput. Phys., 15 (2014), pp. 1407-1430. Correlation Functions, Universal Ratios and Goldstone Mode Singularities in n-Vector Models J. Kaupuzs 1*, R. V. N. Melnik 2, J. Rimsans 11 Institute of Mathematics and Computer Science, University of Latvia, 29 Raina Boulevard, LV1459, Riga, Latvia; Institute of Mathematical Sciences and Information Technologies, University of Liepaja, 14 Liela Street, Liepaja LV-3401, Latvia. 2 Wilfrid Laurier University, Waterloo, Ontario, Canada, N2L 3C5. Received 26 June 2013; Accepted (in revised version) 30 October 2013 Available online 5 March 2014 doi:10.4208/cicp.260613.301013a Abstract Correlation functions in the O(n) models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the O(4) model for lattice sizes about L=120 and small external fields h is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to L=512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at $k \to 0$ and h=+0, i.e., $G_{\perp}(k) \simeq a k^{-\lambda_{\perp}}$ and $G_{\parallel}(k) \simeq b k^{-\lambda_{\parallel}}$, respectively. Here $a$ and $b$ are the amplitudes, $k=|k|$ is the magnitude of the wave vector k. The exponents $\lambda_{\perp}$, $\lambda_{\parallel}$ and the ratio $b M^2/a^2$, where $M$ is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding $b M^2/a^2 = (n-1)/16$. Our MC estimates of this ratio are $0.06 \pm 0.01$ for $n=2$, $0.17 \pm 0.01$ for n=4 and $0.498 \pm 0.010$ for n=10. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for $G_{\perp}(k)$ and $G_{\parallel}(k)$, well fitting the simulation data for small k. We have used them to test the Patashinski-Pokrovski relation and have found that it holds approximately. AMS subject classifications: 65C05, 82B20, 82B80 Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: n-component vector models, correlation functions, Monte Carlo simulation, Goldstone mode singularities. *Corresponding author. Email: kaupuzs@latnet.lv (J. Kaupuzs), rmelnik@wlu.ca (R. Melnik), rimshans@mii.lu.lv (J. Rimsans)