High-precision anomalous dimension of three-dimensional percolation and spatial profile of the critical giant cluster

In three-dimensional percolation, we apply and test the critical geometry approach for bounded critical phenomena based on the fractional Yamabe equation. The method predicts the functional shape of the order parameter profile ϕ, which is obtained by raising the solution of the Yamabe equation to the scaling dimension Δ_{ϕ}. The latter can be fixed from outcomes of numerical simulations, from which we obtain Δ_{ϕ}=0.47846(71) and the corresponding value of the anomalous dimension η=-0.0431(14). The comparison with values of η determined by using scaling relations is discussed. A test of hyperscaling is also performed.