where ˜ v $^{∗}$( k ) = ˜ v $^{T}$( − k ), and The correlation functions for the fluctuations of the solvent composition ϑ and the density of ions η in the Fourier representation are given by the matrix ˜ G = ˜ C − $^{1}$. Each component of this matrix is inversely proportional to det( J − $^{1}$C0 + k $^{2}$I ) = ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$), where λ 2 1 and λ 2 are the two eigenvalues of the matrix and are given by The correlations are thus given by a linear combination of the two terms, ( λ 2 1 + k $^{2}$) − 1 and ( λ 2 + k $^{2}$) − $^{1}$. The asymptotic decay of correlations in real space is described by the inverse correlation length ξ − 1 b = min( λ$_{1}$, λ$_{2}$ ). The uniform phase is stable in the range of parameters T, ¯ s and ¯ ρ$_{c}$ such that det ˜ C ( k ) > 0 for all k < k$_{max}$ , where k$_{max}$ is an upper cutoff on wave numbers | k | ≤ k$_{max}$ = π/a . a is usually identified with some appropriate microscopic length, e.g., the lattice spacing or the molecular diameter. Because det ˜ C ( k ) = J det ( M + k $^{2}$I ) = J ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$) where for the instability analysis we have to distinguish cases of positive and negative J . If J > 0, from the condition det ˜ C (0) ≥ 0 (one-phase stability) we obtain after some algebra that T r M ≥ 0 and λ 2 $_{1}$, λ 2 > 0. In that case the instability of the uniform phase occurs at k = 0 when The condition ˜ C$_{ss}$ = 0 determines the instability for the case of the vanishing field η (the same concentration of ions in the phase-separated solvent). From (34) it follows that λ 2 1 → 0 + so that the asymptotic inverse decay length is 1 /ξ$_{b}$ = λ$_{1}$ . For J < 0, on the other hand, the stability condition of the uniform phase, det ˜ C ( k ) > 0, requires det ( M + k $^{2}$I ) = ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$) < 0 for all k < k$_{max}$ . In particular, the 12 det ˜ C (0) = J det M = Jλ 2 $_{1}$λ 2 = 0 or ˜ C$_{ss}$ = 0 (36) J = det J = J$_{ss}$J$_{ρρ}$ − J 2 $_{sρ}$, (35) λ 2 1 , 2 = T r M ∓ √ ( T r M ) 2 − 4 det M 2 . (34) M = J − $^{1}$C0 (33) ˜ C ( k ) = C 0 + k $^{2}$J = J ( J − $^{1}$C0 + k $^{2}$I ) . (32)