noindent “Integrability meeting ENS/IPhT'' We propose a new method for computing the boundary free energy of an integrable 1+1 dimensional quantum system. We apply the matrix-tree theorem to write the integration measure in the cluster expansion of the partition function as a sum over trees and loops. The sum over trees satisfies the Thermodynamic Bethe Ansatz equation while the sum over loops gives rise to two Fredholm determinants. Based on recent work https://arxiv.org/pdf/1809.05705.pdf (IPhT organizers: Ivan Kostov and Didina Serban)

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