A deletion-invariance property for random measures satisfying the Ghirlanda-Guerra identities
aa r X i v : . [ m a t h . P R ] M a y A deletion-invariance property for random measuressatisfying the Ghirlanda-Guerra identities.
Dmitry Panchenko ∗ December 5, 2018
Abstract
We show that if a discrete random measure on the unit ball of a separable Hilbertspace satisfies the Ghirlanda-Guerra identities then by randomly deleting half of thepoints and renormalizing the weights of the remaining points we obtain the samerandom measure in distribution up to rotations.
Key words: Sherrington-Kirkpatrick model, Gibbs measure.
Let us consider a countable index set A and random probability measure µ on a unit ball B of a separable Hilbert space H such that µ = P α ∈ A w α δ ξ α for some random points ξ α ∈ B and weights ( w α ). We will call indices α from the set A “configuration” and for a function f = f ( α , . . . , α n ) of n configurations we will denote its average with respect to the measure µ by h f i = X α ,...,α n w α · · · w α n f ( α , . . . , α n ) . (1.1)We say that the random measure µ satisfies the Ghirlanda-Guerra identities [3] if for any n ≥ f that depends on the configurations α , . . . , α n only through thescalar products, or overlaps, R ℓ,ℓ ′ = ξ α ℓ · ξ α ℓ ′ for ℓ, ℓ ′ ≤ n we have E h f R p ,n +1 i = 1 n E h f i E h R p , i + 1 n n X ℓ =2 E h f R p ,ℓ i (1.2)for any integer p ≥ . Random measures satisfying the Ghirlanda-Guerra identities arise asthe directing measures (or determinators in the terminology of [8]) of overlap matrices in the ∗ Department of Mathematics, Texas A&M University, email: [email protected]. Partially sup-ported by NSF grant. ε α ) α ∈ A (taking values ± /
2) and for t ∈ R let us define a random measure µ t = P α ∈ A w α,t δ ξ α with weights defined by the random change of density w α,t = w α exp tε α P γ ∈ A w γ exp tε γ . (1.3)and as in (1.1) let us denote the average with respect to this measure by h f i t = X α ,...,α n w α ,t · · · w α n ,t f ( α , . . . , α n ) . (1.4)The following holds. Theorem 1
If a random measure µ satisfies the Ghirlanda-Guerra identities (1.2) then forany t ∈ R , any n ≥ and any bounded function f of the overlaps on n configurations wehave E h f i t = E h f i . The main difference here is that the result holds for all t ∈ R compared to | t | < / t go to infinity we now obtain the following new invariance property. Let η α = ( ε α + 1) / / µ ′ = P α ∈ A w ′ α δ ξ α be the random measure defined by the change of density w ′ α = w α η α P γ ∈ A w γ η γ . (1.5)In other words, we randomly delete half of the point in the support of measure µ andrenormalize the weights to define a probability measure µ ′ on the remaining points. Thedenominator in (1.5) is non-zero with probability one since it is well-known that unless themeasure µ is concentrated at 0 ∈ B (a case we do not consider) it must have infinitely manydifferent points in the support in order to satisfy (1.2). Let us define by h f i ′ the average withrespect to µ ′ . Theorem 2 (Deletion invariance) If a random measure µ satisfies the Ghirlanda-Guerraidentities (1.2) then E h f i ′ = E h f i . emark 1. In particular, this implies that the measure µ ′ also satisfies the Ghirlanda-Guerra identities (1.2) and, hence, we can repeat the random deletion procedure as manytimes as we want. This means that the deletion invariance also holds with random variables( η α ) taking values 1 and 0 with probabilities 1 / s and 1 − / s correspondingly, for anyinteger s ≥ Remark 2.
It is well-known that invariance for the averages as in Theorem 2 implies thatthe random measures µ and µ ′ have the same distribution, up to rotations. Let ( w ℓ ) ℓ ≥ be theweights ( w α ) arranged in the non-increasing order and let ( ξ ℓ ) be the points ( ξ α ) rearrangedaccordingly, so that µ = P ℓ ≥ w ℓ δ ξ ℓ . Similarly, let µ ′ = P ℓ ≥ w ′ ℓ δ ξ ′ ℓ . Then arguing as at theend of the proof of Theorem 4 in [4] (or Lemma 4 in [5]) one can show that (cid:0) ( w ℓ ) ℓ ≥ , ( ξ ℓ · ξ ℓ ′ ) ℓ,ℓ ′ ≥ (cid:1) d = (cid:0) ( w ′ ℓ ) ℓ ≥ , ( ξ ′ ℓ · ξ ′ ℓ ′ ) ℓ,ℓ ′ ≥ (cid:1) (1.6)which means that up to rotations the configurations of the random measures µ and µ ′ havethe same distributions. Proof of Theorem 1.
Suppose that | f | ≤ . Let ϕ ( t ) = E h f i t and by symmetry we onlyneed to consider t ≥
0. Given configurations α , α , . . . let us denote D n = ε α + . . . + ε α n − nε α n +1 and a straightforward computation shows that ϕ ′ ( t ) = E h f D n i t and similarly for all k ≥ ,ϕ ( k ) ( t ) = E h f D n · · · D n + k − i t . It was proved in Theorem 4 in [4] (a more streamlined proof was given in Theorem 6.3 in[6]) that if the measure µ satisfies the Ghirlanda-Guerra identities (1.2) then ϕ ( k ) (0) = 0 for all k ≥ . (1.7)It is also easy to see that | D n | ≤ n so that for all t, | ϕ ( k ) ( t ) | ≤ k n ( n + 1) · · · ( n + k − . (1.8)This is all one needs to show that if ϕ ( t ) = ϕ (0) and ϕ ( k ) ( t ) = 0 for all k ≥ t ≤ t for some t ≥ t < t + 1 / . This will finish theproof of the theorem since by (1.7) this holds for t = 0. Take any k ≥ . Using (1.8) and(1.9) for t = t and using Taylor’s expansion for a function ϕ ( k ) ( t ) around the point t = t we get for any m ≥ | ϕ ( k ) ( t ) − ϕ ( k ) ( t ) | ≤ sup t ≤ s ≤ t | ϕ ( k + m ) ( s ) | m ! | t − t | m ≤ k + m n ( n + 1) · · · ( n + k + m − m ! | t − t | m . If | t − t | < / m → ∞ proves that ϕ ( k ) ( t ) = ϕ ( k ) ( t ) for all k ≥ t < t + 1 /
2. 3 roof of Theorem 2.
Let I = { α ∈ A : ε α = 1 } and let Z t = X α ∈ I w α + e − t X α ∈ I c w α so that w α,t = w α Z t (cid:0) I ( α ∈ I ) + e − t I ( α ∈ I c ) (cid:1) . (1.10)Then the sum on the right hand side of (1.4) can by broken into 2 n groups depending onwhich of the indexes α , . . . , α n belong to I or its complement I c , for example, the termscorresponding to all indices belonging to I will give1 Z nt X α ,...,α n ∈ I w α · · · w α n f ( α , . . . , α n ) . (1.11)This sum is bounded by one and when t → + ∞ it obviously converges to h f i ′ while thesums corresponding to other groups, when at least one of the indices belongs to I c , willconverge to zero because of the factor e − t in (1.10). By dominated convergence theorem weget convergence of expectations. References [1] Aizenman, M., Contucci, P. (1998) On the stability of the quenched state in mean-fieldspin-glass models.