Spacelike thermal correlators are almost time-independent

We show that, in relativistic field theories, the thermal correlation function of N bosonic operators <O_1(x_1,t_1) O_2(x_2,t_2) … O_N(x_N,t_N)> at sufficiently spacelike-separated points shows exponentially weak dependence on the time variables t_1,…,t_N, when the space separations are held fixed. For classical thermal field theory, the time dependence vanishes when all points are spacelike separated