  We considered random discrete approximation of O'Hara energy. O'Hara energy
is the energy defined for a knot, and O'Hara energy was introduced for defining
the standard shape for each knot class (equivalence class by ambient isotopy)
by variational method. In the case of a specific exponent, due to energy
invariance under Moebius transformation, this energy is called Moebius energy.
Although discretization for various Moebius energies has been defined to
analyse the shape of the minimizer so far, only Gamma-convergence to the
original energy has been shown for a conventional discretization. In this
study, we are successful to show locally uniform convergence and compactness of
discrete energy in a space based on optimal transport theory, by introducing
random discrete approximation of O'Hara energy using random variable.
