Systems of free fermions trapped under a certain class of potentials and at zero temperature are known to be related to the GUE matrix model and so their limiting behaviors are described by the Airy and sine kernels and their determinantal point processes. When instead these systems are considered at non-zero temperature, some ''finite temperature'' deformations of the Airy and sine kernels appear. In this talk we are going to compare the analytical properties of some spacing distributions in the corresponding determinantal point processes. In particular we will see that they still satisfy some "Tracy-Widom formula", generalizing the well-known ones of the classical cases, and they are related to certain solutions of integrable partial differential equations.
