Exact scattering matrix of graphs in magnetic field and quantum noise

We consider arbitrary quantum wire networks modelled by finite, noncompact,connected quantum graphs in the presence of an external magnetic field. We find a general formula for the total scattering matrix of the network interms of its local scattering properties and its metric structure. This isapplied to a quantum ring with $N$ external edges. Connecting the externaledges of the ring to heat reservoirs, we study the quantum transport on thegraph in ambient magnetic field. We consider two types of dynamics on the ring:the free Schr\ödinger and the free massless Dirac equations. For each case, adetailed study of the thermal noise is performed analytically. Interestingly enough, in presence of a magnetic field, the standard linearJohnson-Nyquist law for the low temperature behaviour of the thermal noisebecomes nonlinear. The precise regime of validity of this effect is discussedand a typical signature of the underlying dynamics is observed.