Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation

We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a $2+1$-dimensional spacetime, extending the work of Moretti and Pinamonti in a $3+1$-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the $3+1$-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state exists in a neighbourhood of the Killing horizon, we show that this is not an empty condition by verifying it for a massive, real scalar field subject to Robin boundary conditions in the prototypic example of a three dimensional black hole background: the non-extremal, rotating BTZ spacetime.