# Mathematics Colloquia and Seminars

In recent years the thermal $P(\phi)_2$-models were established in full rigour, and it is now possible to take a more detailed look at their properties. It was shown that the correlation functions of the thermal $P(\phi)_2$-models decay exponentially in space-like directions. We conjecture that the exponential decay of the correlation functions of the thermal $P(\phi)_2$-model implies the existence of a mass gap in the K\"all\'en-Lehmann decomposition of the thermal two-point function. This result is relevant in the context of spontaneous symmetry breaking at positive temperature. We will also briefly comment on situation in the charged $P(\phi)_2$-model with non-vanishing chemical potential.