  We analytically estimate the locations of phase transition points in the
ground state for the $\pm J$ random bond Ising model with asymmetric bond
distributions on the square lattice. We propose and study the percolation
transitions for two types of bond shared by two non-frustrated plaquettes. The
present method indirectly treats the sizes of clusters of correlated spins for
the ferromagnetic and spin glass orders. We find two transition points. The
first transition point is the phase transition point for the ferromagnetic
order, and the location is obtained as $p_c^{(1)} \approx 0.895 \, 399 \, 54$
as the solution of $[p^2 + 3 (1-p)^2 ]^2 \, p^3 - \frac{1}{2} = 0$. The second
transition point is the phase transition point for the spin glass order, and
the location is obtained as $p_c^{(2)} = \frac{1}{4} [2 + \sqrt{2 (\sqrt{5} -
1)}] \approx 0.893 \, 075 \, 69$. Here, $p$ is the ferromagnetic bond
concentration, and $1 - p$ is the antiferromagnetic bond concentration. The
obtained locations are reasonably close to the previously estimated locations.
This study suggests the presence of the intermediate phase between $p_c^{(1)}$
and $p_c^{(2)}$; however, since the present method produces remarkable values
but has no mathematical proof for accuracy yet, no conclusions are drawn in
this article about the presence of the intermediate phase.
