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In this paper we present the proof of the convergence of the critical bondpercolation exploration process on the square lattice to the trace ofSLE$_{6}$. This is an important conjecture in mathematical physics andprobability. The case of critical site percolation on the hexagonal lattice wasestablished in the seminal work of Smirnov via proving Cardy's formula. Ourproof uses a series of transformations and conditioning to construct a pair ofpaths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the sitepercolation case on the hexagonal lattice allows us to obtain certain estimateson the scaling limit of the $+\partial$CBP and the $-\partial$CBP. Byconsidering a path which is the concatenation of $+\partial$CBPs and$-\partial$CBPs in an alternating manner, we can prove the convergence in thecase of bond percolation on the square lattice.

Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice