Journal of Statistical Physics

, Volume 20, Issue 6, pp 573–584 | Cite as

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

  • K. M. Khanin
  • Ya. G. Sinai
Articles

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(x1,x2) are independent random variables for different pairs (x1,x2),(x1,x2) = 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.

Key words

Random interactions random variables long range free energy Hamiltonian spin system partial function 

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Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • K. M. Khanin
    • 1
  • Ya. G. Sinai
    • 1
  1. 1.L. D. Landau Institute for Theoretical PhysicsThe Academy of Sciences of the USSRMoscow

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