We obtain a subgroup of the isometry group of pAdS (a p-adic version of AdS alternative to the Bruhat-Tits tree). We propose a candidate for the scalar bulk action and equation of motion on pAdS, and work out analytical expressions of the Green's functions for a particular choice of parameter together with an ansatz for general cases. The limiting behaviors of the Green's function are also studied. With their help, the convergence of small loops (whose radii are smaller than AdS length scale of pAdS) is analyzed. * qufeng@itp.ac.cn

For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary theory is very similar to a scalar field theory over p-adic numbers. * qufeng@itp.ac.cn

The Schwinger pairs production rate is calculated numerically in the soft-wall model with the help of a simpler method in determining the soft-wall's position beyond which probe strings connecting the Schwinger pairs do not fall into. The critical behaviour of the production rate and linear part in the middle region are both studied carefully. The latter manifests interesting new features. The results are compared with those in previous hard-wall models.