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.