Mesoscopic critical fluctuations

We investigate magnetic fluctuations of a mesoscopic critical region at the interface induced by smooth time-independent spacial changes of a control parameter across its critical value. Near the spatial critical point, the order parameter fluctuations are mesoscopic, i.e., much larger than the lattice constant but decaying away from the critical region. We propose a minimal model to describe this behavior and show that it leads to the integrable Painlev\'e-II equation for the local order parameter. We argue that known mathematical properties of this equation produce insight into the nonlinear susceptibilities of this region.