The lace expansion is a powerful tool to analyze the critical behavior in high dimension. It is applied to various stochastic models, such as self-avoiding walk, percolation, Ising model etc. It is conjectured that the nearest-neighbor ordinary and oriented percolations exhibit the mean-field behavior in dimension $d$ above the upper critical dimension $d_\mathrm{c}=6, 4$, respectively, so that we want to prove them by the lace expansions. To achieve this, we consider the $d$-dimensional body-centered cubic (BCC) lattice.

In this talk, I will show current statuses of the above problems. The research on the ordinary percolation is based on a joint works with Satoshi Handa (Fujitsu Laboratories Ltd.) and Akira Sakai (Hokkaido University, Japan). That on the oriented percolation is based on a joint work with Satoshi Handa (id.) and Lung-Chi Chen (National Chengchi University, Taiwan).