The famous Tracy-Widom (T-W) law governs the fluctuations of the largest eigenvalue of a random hermitian matrix. Its appearance is not limited to random matrix theory: since its discovery, it has been found to describe, after non trivial rescaling, the edge behavior of several combinatorial problems, out of equilibrium exclusion processes such as ASEP, or 'arctic circle' type problems in equilibrium statistical mechanics. This distribution can be understood --from a physicist perspective-- using a simple 1d free fermions model in a linear potential, and often pops up in setups that are free fermions in disguise. In this talk I will discuss a few examples of interacting quantum systems with inhomogeneous couplings were T-W naturally appears in the ground state, and give a simple argument why this occurs. In the case of Lieb-Liniger models in trapping potentials or XXZ spin chain in varying magnetic field, Bethe Ansatz considerations allow us to compute the appropriate scaling exactly. If time permits I will also discuss interacting quantum out of equilibrium setups where T-W scaling typically breaks down.
