Correlation Length Bounds for Disordered Ising Ferromagnets

The d-dimensional, nearest-neighbor disordered Ising ferromagnet: is studied as a function of both temperature, T, and a disorder parameter, λ, which measures the size of fluctuations of couplings J^ ^O. A finite-size scaling correlation length, ξf(T, λ), is defined in terms of the magnetic response of finite samples. This correlation length is shown to be equivalent, in the scaling sense, to the quenched average correlation length ξ(T,λ)9 defined as the asymptotic decay rate of the quenched average two-point function. Furthermore, the magnetic response criterion which defines ξf is shown to have a scale-invariant property at the critical point. The above results enable us to prove that the quenched correlation length satisfies: C\logξ(T)\ξ(T)^\T-TcΓ , which implies the bound v ̂ 2/d for the quenched correlation length exponent.