  The conventional duality analysis is employed to identify a location of a
critical point on a uniform lattice without any disorder in its structure. In
the present study, we deal with the random planar lattice, which consists of
the randomized structure based on the square lattice. We introduce the
uniformly random modification by the bond dilution and contraction on a part of
the unit square. The random planar lattice includes the triangular and
hexagonal lattices in extreme cases of a parameter to control the structure.
The duality analysis in a modern fashion with real-space renormalization is
found to be available for estimating the location of the critical points with
wide range of the randomness parameter. As a simple testbed, we demonstrate
that our method indeed gives several critical points for the cases of the Ising
and Potts models, and the bond-percolation thresholds on the random planar
lattice. Our method leads to not only such an extension of the duality analyses
on the classical statistical mechanics but also a fascinating result associated
with optimal error thresholds for a class of quantum error correction code, the
surface code on the random planar lattice, which known as a skillful technique
to protect the quantum state.

Disordered Systems and Neural Networks;Statistical Mechanics;Quantum Physics