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 Γ-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, and we can show convergence from the minimizer to
the minimizer.

This is a joint seminar with NLPDE seminar.