Fredholm Determinants from Schr\"odinger Type Equations, and Deformation of Tracy-Widom Distribution

We undertake an analysis of Fredholm determinants arising from kernels whose defining functions satisfy a Schr\"odinger type equation. When this defining function is the Airy one, the evaluation of the corresponding Fredholm determinant yields the notorious Tracy-Widom distribution [hep-th/9211141], which has found many applications in numerous domains. In this paper, we unveil a generalization of the Tracy-Widom distribution for a generic class of defining functions. Furthermore, we bring forth a direct application of our upshot and survey the relation between the framework which we employ and isomonodromic systems.