For general 2d-random ergodic magnetic Schrödinger operators, we establish a direct connection between bulk-edge correspondence and the theory of diamagnetic currents and of Landau diamagnetism.
Our main result is encapsulated in a formula which states that the derivative of a large class of bulk partition functions with respect to the external constant magnetic field, equals the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Neither spectral gaps, nor mobility gaps, nor topological arguments are required.
The equality between the bulk and edge indices, as stated by the conventional bulk-boundary correspondence, is obtained as a corollary of our purely analytical arguments by imposing a gap condition and by taking a "zero temperature" limit. This is joint work with M. Moscolari and S. Teufel.