We will discuss a method to calculate the rate of false vacuum decay induced by a black hole. The method uses complex tunnelling solutions and consistently takes into account the structure of quantum vacuum associated to the black hole. We will illustrate the technique on a two- dimensional toy model of a scalar field with inverted Liouville potential in an external background of a dilaton black hole. Using this model, we will compute the exponential suppression of tunnelling from the Boulware, Hartle-Hawking and Unruh vacuum states and show that they are parametrically different. Finally, we will discuss how our results are generalised to the realistic case of black holes in four dimensions.