Download Communications in Mathematical Physics - Volume 239 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Extra resources for Communications in Mathematical Physics - Volume 239

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The νdimensional Lebesgue measure, leads to ϕˆσ ν,b = ϕˆ1 ν,bσ . Similarly ε d ϕˆσ ν,b = εσ d ϕˆ1 ν,bσ . Concentration Inequalities 39 anymore but only on the Sobolev norm of its differential. Next c, ct have to be taken as constants for the ω-distribution for that particular η. An equal game can be played by exchanging the roles of η and ω, so that we are fixing the latter ones. Note that, when ω is fixed we are left with a model on a distorted but fixed point set {x + ωx , x ∈ } (with modified but positive minimal packing radius b/2).

Fact about Dobrushin uniqueness. Suppose that is a countable set, infinite or finite, and the random variables (Xx )x∈ are distributed according to a Gibbs measure ρ that n obeys the Dobrushin uniqueness condition (see the Introduction). Put D = ∞ n=0 C , where C is the interdependence matrix of ρ. 3) ξ with constants bx for x ∈ . Then the expectations of any function f (ξ ) on the infinitevolume configurations ξ don’t differ more than |ρ(f ) − ρ(f ˜ )| ≤ δy (f )Dy,x bx . 4) y,x∈ To show Lemma 1 let us use short notations like µ(F (X)|Tx )(ξ ) ≡ µ(F (X) ξ≥x ), etc.

3). Collecting terms Theorem 3 follows. A different (although less natural) way to prove the “total concentration result” of Theorem 3 would be to prove that the joint distribution can be represented as a Gibbsmeasure for the joint variables ξx = (ηx ωx ), estimate its joint constants c, ct , and then apply Theorem 4. Note in this context that it won’t be true in general that the resulting measure is a Gibbs measure, even for independent Xx ’s, when one allows for conditional Gibbsian distributions of the Y -variables having phase transitions (which is however excluded here).

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