  We report on the exact treatment of a random-matrix representation of bond
percolation model on a square lattice in two dimensions with occupation
probability $p$. The percolation problem is mapped onto a random complex matrix
composed of two random real-valued matrices of elements $+1$ and $-1$ with
probability $p$ and $1-p$, respectively. We find that the onset of percolation
transition can be detected by the emergence of power-law divergences due to the
coalescence of the first two extreme eigenvalues in the thermodynamic limit. We
develop a universal finite-size scaling law that fully characterizes the
scaling behavior of the extreme eigenvalue's fluctuation in terms of a set of
universal scaling exponents and amplitudes. We make use of the relative entropy
as an index of the disparity between two distributions of the first and
second-largest extreme eigenvalues, to show that its minimum underlies the
scaling framework. Our study may provide an inroad for developing new methods
and algorithms with diverse applications in machine learning, complex systems,
and statistical physics.
