Multiple Ginzburg-Landau vortices pinned by randomly distributed small holes

In this work a minimization problem for the magnetic Ginzburg-Landau functional in a circular domain with randomly distributed small holes is considered. We develop a new analytical approach for solving a non-standard boundary value problem for the magnetic field presented as a function of the n-tuple of degrees of vortices pinned by the n holes. The key feature of this approach is that the solution is analytically derived via the method of functional equations and does not rely on periodic geometry as in previous studies. We prove the convergence of the method of successive approximations applied to the functional equations for arbitrary hole locations. Once established the associated energy functional is minimized as a function of vortex degrees to find the effective vorticity distribution. After verification of the method by comparison of our results to previous works in the periodic case, we identify the striking differences due to the presence of random hole locations. Namely, the different subdomain structure, the proximity of hole vortices to the boundary and our approach allows for the estimation of the fractal dimension of the interface between regions of like degree.