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By using the determinant Quantum Monte Carlo method, the magnetic and pairingcorrelation of the Na$_{x}$CoO$_{2}\cdot$yH$_{2}$O system are studied withinthe Hubbard model on a bilayer triangular lattice. The temperature dependenceof spin correlation function and pairing susceptibility with several kinds ofsymmetries at different electron fillings and inter layer coupling terms areinvestigated. It is found that the system shows an antiferromagneticcorrelation around the half filling, and the $fn$-wave pairing correlationdominates over other kinds of pairing symmetry in the low doping region. As theelectron filling decreases away from the half filling, both the ferromagneticcorrelation and the $f$-wave paring susceptibility are enhanced and tend todominate. It is also shown that both the magnetic susceptibility and paringsusceptibility decrease as the inter layer coupling increases.

Numerical study of magnetic and pairing correlation in bilayer triangular lattice