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By M. Aizenman (Chief Editor)

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Extra resources for Communications in Mathematical Physics - Volume 224

Example text

If we set U (q) = ∪|Q|=q MQ the phase diagram satisfies the Gibbs phase rule, provided there are exactly |Q| tangent functionals at H u + T for each u ∈ MQ . β,u,T Each metastable free energy fj , j ∈ Q, defines a tangent functional αj : for all β,u,T +ηK ∂ fj |η=0 . Notice that item (c) ensures boundedness K ∈ Br , we set αj (K) = ∂η 2 of the tangent functional. We show now that these tangent functionals are linearly independent, and that any other tangent functional is a linear combination of these ones.

We denote by = {1, . . , M}Z the space of classical configurations; the dimension ν of the physical space is always supposed to be bigger or equal to 2. The interaction has the form H = V +T , where V is a block interaction and is diagonal with 48 J. Fröhlich, L. Rey-Bellet, D. 1) and VA = 0 if there is no x with U (x) = A. The function x depends on µ ∈ U ⊂ Rp−1 , and we assume that its derivatives ∂µ∂ j x (ωU (x) ) are bounded uniformly in x, µ, ω, j . A finite set G = {g (1) , . . , g (p) } ⊂ of periodic configurations is given, that contains all ground states of V for all µ (see below the precise assumption).

E. we have w β,u,T (τa A) = w β,u,T (A) for all a ∈ ( Z)ν and all A. Here τa is the translation operator. u e−βe0 |A| e−τ |A| for a large enough constant τ (depending on ν, p, • |w β,u,T (A)| and ). Furthermore, | ∂ β,u,T w (A)| ∂ui u β|A|C e−βe0 |A| e−τ |A| and | ∂ β,u,T +ηK (A)| w ∂η β|A|C K r u e−βe0 |A| e−τ |A| for a uniform constant C. • limβ→∞ limT →0 w β,u,T (A) = 0. This means that the weights represent the correction to the situation (β = ∞, T = 0). • wβ,u,T (A) is real analytic in u; for all K ∈ Br , wβ,u,T +ηK (A) is real analytic in η in a neighborhood of 0 (the neighborhood depends on K).

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