Time-dependent temperature correlators of local spins of the one-dimensional Heisenberg $XY$ chain

Time-dependent temperature correlators of the anisotropic Heisenberg $XY$ chain are calculated by making use of technique of integration over Grassmann variables. For the chain of length $M$ the correlators are represented as determinants of $2M\times2M$ matrices. In the thermodynamic limit the correlation functions are expressed in terms of the Fredholm determinants of linear integral operators with a matrix kernel.