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Hawking effect for a toy model of interacting fermions

We consider a toy model of interacting Dirac fermions in a 1+1 dimensional space time describing the exterior of a star collapsing to a blackhole. In this situation we give a rigorous proof of the {\em Hawking effect}, namely that under the associated quantum evolution, an initial vacuum state will converge when $t\to +\infty$ to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null characteristics, and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.