 Percolation triangle condition

Here $\tau (x,y)=P_{p_c}(x\leftrightarrow y)$ is critical two-point connectivity. The condition controls the intersection effects that spoil a branching approximation. It is a sufficient criterion for important mean-field percolation exponents, including the order-parameter, percolation susceptibility and cluster-tail exponents. Establishing detailed spatial asymptotics is a further task, commonly addressed by lace expansion.