Level spacing functions and connection formulas for Painlevé V transcendent

The connection formulas for Painlevé V transcendents are applied for calculations of the Fredholm determinant of the kernel sin π(x−y)/π(x−y) on the finite interval (t,−t). The level spacing functions, arising in the theory of random matrices as well as for the correlators of one-dimensional XY-model, are expressed via this Fredholm determinant. Its asymptotics for large t come from nonlinear second-order ODE with known initial conditions at t=0. The Bäcklund transformations reduce this ODE to the Painlevé V equation, while the isomonodromic deformation method (IDM) provides the explicit connection formulas for the necessary PV transcendent. The result generalizes the well-known Dyson asymptotic expansion for the one-level spacing function to a case of arbitrary many levels.