  In this paper, we construct an adiabatic invariant for a large 1--$d$ lattice
of particles, which is the so called Klein Gordon lattice. The time evolution
of such a quantity is bounded by a stretched exponential as the perturbation
parameters tend to zero. At variance with the results available in the
literature, our result holds uniformly in the thermodynamic limit. The proof
consists of two steps: first, one uses techniques of Hamiltonian perturbation
theory to construct a formal adiabatic invariant; second, one uses
probabilistic methods to show that, with large probability, the adiabatic
invariant is approximately constant. As a corollary, we can give a bound from
below to the relaxation time for the considered system, through estimates on
the autocorrelation of the adiabatic invariant.
