Critical Behavior of a Semi-infinite System:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>-Vector Model in the Large-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>n</mml:mi></mml:math>Limit

The $n$-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at $T={T}_{c}$ in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In the scaling regime the spin-spin correlation function is $G(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}, z, {z}^{\ensuremath{'}}, {T}_{c})=\mathrm{const}{{{[{\ensuremath{\rho}}^{2}+{(z\ensuremath{-}{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}\ensuremath{-}{[{\ensuremath{\rho}}^{2}+{(z+{z}^{\ensuremath{'}})}^{2}]}^{\ensuremath{-}1}}}^{\frac{(d\ensuremath{-}2)}{2}}$, for dimensionalities $d$ in the range $2&lt;d&lt;4$, where $z$, ${z}^{\ensuremath{'}}$ are the distances of the two spins from the surface and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ is their separation parallel to the surface. The critical exponents ${\ensuremath{\eta}}_{\ensuremath{\perp}}$ and ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$ are $\frac{(d\ensuremath{-}2)}{2}$ and ($d\ensuremath{-}2$), respectively.