Existence of free energy for models with long-range random Hamiltonians

Abstract

Classical lattice systems with random Hamiltonians $$\frac{1}{2}\sum\limits_{x_1 \ne x_2 } {\frac{{\varepsilon (x_1 ,x_2 )\varphi (x_1 )\varphi (x_2 )}}{{\left| {x_1 - x_2 } \right|^{\alpha d} }}}$$ are considered, whered is the dimension, andε(x 1,x 2) are independent random variables for different pairs (x 1,x 2),Eε(x 1,x 2) = 0. It is shown that the free energy for such a system exiists with probability 1 and does not depend on the boundary conditions, providedα > 1/2.