Log-Sobolev inequalities for unbounded lattice spin systems^{1}

Nobuo Yoshida^{2}

The purpose of this talk is to present a result on the log-Sobolev inequality for the ϕ^{4}-lattice field at high temparature. The talk will be organized as follows;

1. The result and the background

2. The first half of the proof (“High temparature” ⇒ “mixing condition”)

3. The second half of the proof (“mixing condition” ⇒ log-Sobolev)

I will explain what exactly the mixing condition is later, which is, unfortunately not easy to write down. But good thing about the mixing condition is that it is mild enough to be checked and is at the same time, strong enough to prove log-Sobolev with.

1 The result and the background

We consider a spin system on Z^{d} with the following Hamiltonian; for Λ ⊂⊂ Z^{d} and its boundary condition ω ∈ R^{Λ^{c}}, we define a function H^{Λ,ω} : R^{Λ} ⟶ R by

H^{Λ,ω}(σ) = − ∑_{x,y∈Λ} J_{x,y}σ_{x}σ_{y} − 2 ∑_{{x∈Λ \atop y∉Λ}} J_{x,y}σ_{x}ω_{y} + ∑_{x∈Λ} U_{a,b}(σ_{x}),  (1.1)

where

σ = (σ_{x})_{x∈Λ} ∈ R^{Λ}
J_{x,y} = JΔ_{x,y} − mδ_{x,y}

J > 0, m > 0, and Δ denotes the Laplacian on Z^{d}

U_{a,b}(s) = as^{4} − bs^{2},

a > 0, b ∈ R.

The finite volume Gibbs state is then defined by;

E^{Λ,ω}(dσ) = \frac{exp −H^{Λ,ω}(σ)}{normalization} ∏_{x∈Λ} dσ_{x}  (1.2)

The main result I would like to present in this talk is the following:

Theorem 1.1 ([Y97]) Let d ≥ 1, m > 0, a > 0 and b ∈ R be arbitrary. Then, for sufficiently small J > 0, there is γ > 0 such that

E^{Λ,ω} (f^{2} log \frac{f^{2}}{E^{Λ,ω}(f^{2})}) ≤ γE^{Λ,ω} (∑_{x∈Λ} |\frac{∂f}{∂σ_{x}} |^{2}) ,  (1.3)

for all Λ ⊂⊂ Z^{d}, ω ∈ R^{Λ^{c}}, f ∈ C_{b}^{1}(R^{Λ} → R).

1A note for a seminar talk on Jun 5, 1998 at IHP, Paris

2[e-mail]nobuo@kusm.kyoto-u.ac.jp [URL]http://www.kusm.kyoto-u.ac.jp /ñobuo

Remark 1.1 (1.3) implies the uniqueness of tempered Gibbs state, where a Borel probability measure μ on R^{Z^{d}} is called a tempered Gibbs state if the following two conditions are satisfied;

DLR-equation; ∫ μ(R^{Λ} × dω_{Λ^{c}})E^{Λ,ω}(dσ_{Λ}) = μ(dσ_{Λ} × dω_{Λ^{c}}), ∀Λ ⊂⊂ Z^{d}.

tempered; ∃p > 0, lim_{|x|↗∞} |x|^{−p}μ(|σ_{x}|) = 0.

Remark 1.2 Recently, log-Sobolev inequalities for unbounded spin models are studied under various conditions.

(1) m > b ( [AKR95]); In this case the Hamiltonian is a strictly convex function in that

(\frac{∂^{2}H^{Λ,ω}}{∂σ_{x}∂σ_{y}} (σ))_{x,y∈Λ} ≥ m − b > 0.  (1.4)

Therefore, the Bakry-Emery criterion ([BE85]) for log-Sobolev applies.

(2) d = 1 and a, b and m are arbitrary ([Z96, Theorem 4.1]); The B-E criterion is no longer available in this case. Using the geometrical construction instead, one proceeds as follows;

d = 1 ⇒ “a very strong mixing condition” ⇒ log-Sob.

(See [Z96, Lemma 4.5] for the precise statement of the “very strong mixing condition”. ) It should also be mentioned that [Z96] contains another interesting result which says for any d ≥ 1 that

a decay of correlation ⇒ log-Sob. for ω ≡ 0,  (1.5)

where the exact meaning of “decay of correlation” is that there is C_{1.6} ∈ (0, ∞) such that

|E^{Λ,ω}(σ_{x}; σ_{y})| ≤ C_{1.6} exp (−\frac{d(x, y)}{C_{1.6}}) ,  (1.6)

for any Λ ∈ F, ω ∈ R^{Λ^{c}} and x, y ∈ Λ.

(3) J << 1 and a, b and m are arbitrary([Y97, He97]); A few weeks after the first version of [Y97] was completed, I received a paper of B. Helffer [He97] in which the following result is obtained:

J << 1 ⇒ (1.6) and hence log-Sob for ω ≡ 0

Helffer’s paper deals with more general potential function (not necessarily a polynomial). We will come back to this point later (cf. Remark 2.1 below).

2 “High temparature” ⇒ “mixing condition”

The mixing condition: To write down the mixing condition, we introduce the following new measure. For Δ ⊂ Λ ⊂⊂ Z^{d}, ω ∈ R^{Λ^{c}}, \overline{p}, q ∈ R^{Λ}, we define a measure E_{Λ,q}^{Δ,\overline{p}}(dp_{Δ}) on R^{Δ} by

E_{Λ,q}^{Δ,\overline{p}}(dp_{Δ})
= E^{Λ,ω} ⊗ E^{Λ,ω} \{(σ^{1}, σ^{2}) ; (\frac{σ_{x}^{1} − σ_{x}^{2}}{\sqrt{2}} )_{x∈Δ} ∈ dp_{Δ} |\frac{σ^{1} − σ^{2}}{\sqrt{2}} = \overline{p} on Λ\Δ, \frac{σ^{1} + σ^{2}}{\sqrt{2}} = q, \}

The mixing condition is then stated as follows; There is C ∈ (0, ∞) such that

sup_{q} |E_{Λ,q}^{Δ,\overline{p}^{1}}(p_{z}) − E_{Λ,q}^{Δ,\overline{p}^{2}}(p_{z})|

≤ C (1 + ∑_{w∈Λ∩∂Δ}(|\overline{p}_{w}^{1}| + |\overline{p}_{w}^{2}|)) exp (−\frac{d(y, z)}{C})  (2.1)

whenever z ∈ Δ ⊂ Λ, y ∈ Λ\Δ, \overline{p}^{1} ≡ \overline{p}^{2} off y.

The first half of the proof of Theorem 1.1 can be summerized as follows;

J << 1
⇒^{1} E_{Λ,q}^{Δ,\overline{p}} satisfies the “Dobrushin-Shlosman constructive critrion”
⇒^{2} mixing condition (2.1)

The “constructive critrion” is a stronger version of the original one stated in [DSh85, Theorem 2.1]. Ideas from [COPP78, proof of Theorem 2.3] and [AKRT97, proof of Theorem 4.1] are employed to prove “ ⇒^{1} ”. The proof of “ ⇒^{2} ” is to carefully accommodate the coupling arguments in [DSh85, proof of Theorem 2.1] to our framework.

Remark 2.1 It is not difficult to prove that (2.1) implies (1.6) (cf. [Y97, Lemma 3.4]). On the other hand, the proof of “ ⇒^{1} ” and “ ⇒^{2} ” work for potential function considered in [He97]. Therefore, we recover the result in [He97] at this point.

3 “mixing condition” ⇒ log-Sobolev

The second half of the proof of Theorem 1.1 can be summerized as follows;

mixing condition (2.1)

⇒^{3} estimate of the commutator [E^{Λ}, ∇]
⇒^{4} log-Sobolev

The precise statement of the “commutator estimate” can be found in [Y97, Lemma 3.5]. Proof of “⇒^{3}” is a generalization of the arguments in [Z96, proof of Theorem 4.1], whereas “⇒^{4}” can be carried out by “recursion by doubling” argument ([LY93]).

References

[AKR95] Albeverio, S., Kondratiev, Yu, G. and Röckner, M. : Dirichlet operators via stochastic analysis, J. Funct. Anal. 128, 102–138, (1995).

[AKRT97] Albeverio, S., Kondratiev, Yu, G. and Röckner, M. and Tsikalenko, T. V.:x Uniqueness of Gibbs states for quantum lattice systems, Prob.Th.Rel.Fields. 108,193–218, (1997) .

[BE85] Bakry, D. and Emery, M.: Diffusions hypercontractives, Séminaire de Probabilités XIX, Springer Lecture Notes in Math., 1123, 177–206, (1985).

[BH82] Bellissard, J. and Hϕegh-Krohn, R.: Compactness and maximal Gibbs state for random Gibbs fields on the lattice, Commun. Math. Phys. 84,297–327, (1982).

[COPP78] Cassandro, M., Olivieri, E., Pellegriotti, A. and Presutti, E.: Existence and uniqueness of DLR measures for unbounded spin systems, Z. Warsch. verw. Gebiete 41, 313–334, (1978).

[DSh85] Dobrushin, R. L. and Shlosman, S. : Constructive criterion for the uniqueness of Gibbs field, in “Statistical Physics and Dynamical Systems” ed. by J. Fritz, A. Jaffe and D. Szasz, Birkhäuser (1985).

[He97] Helffer, B. Remarks on the decay of correlations and Witten Laplacians III–Application to logarithmic Sobolev inequalities.

[LY93] Lu, S. L. and Yau, H. T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Commun. Math. Phys. 156, 399–433, (1993).

[SZ92] Stroock, D. W. and Zegarlinski, B. : The logarithmic Sobolev inequality for discrete spin systems on the lattice, Commun. Math. Phys. 149, 175-193 (1992).

[Y97] Yoshida, N.: The log-Sobolev inequality for weakly coupled ϕ^{4}-lattice fields, to appear in Probab. Th. and Rel. Field (1997).

[Z96] Zegarlinski, B. : The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175, 401–432, (1996).