Skeleton inequalities and mean field properties for lattice spin systems Available to Purchase

D.

Brydges

,

J.

Fröhlich

, and

A. D.

Sokal

, , ().

D.

Brydges

,

J.

Fröhlich

, and

A. D.

Sokal

, , ().

A. D.

Sokal

, , ().

M.

Aizenman

, , ().

M.

Aizenman

, , ().

M.

Aizenman

and

R.

Graham

, [FS9], ().

C.

Aragão De Carvalho

,

S.

Caracciolo

, and

J.

Fröhlich

, [FS7], ().

J.

Fröhlich

, [FS4], ().

T.

Hattori

, , ().

A.

Bovier

and

G.

Felder

, , ().

The inequalities are also useful for the construction of continuum field theories in three dimensions with bare Lagrangian containing terms like $ρ2nφ2n.$ Here the bare couplings $ρ2n$ are adjusted to order $εn−2$ where ε is the lattice spacing. (For the details, see T. Hara, “Construction of a Nontrivial Field Theory in 3 dimensions Starting from a Lagrangian of $φ6$ type” University of Tokyo‐Komaba 84‐15).

D. Ruelle, Statistical Mechanics—Rigorous Results (Benjamin, New York, 1969).

D. Brydges, “Field theories and Symanzik’s polymer representation,” in Gauge Theories—Fundamental Interactions and Rigorous Results (Proceedings of the 1981 Poiana Brasov Summer School) edited by G. Dita, V. Georgescu, and R. Purice (Birkhäuser, Boston, 1982).

D.

Brydges

,

J.

Fröhlich

, and

T.

Spencer

, , ().

F.

Dunlop

and

C. M.

Newman

, , ().

J.

Ginibre

, , ().

R. B.

Griffiths

, , , ().

G. S.

Sylvester

, , ().

Here, and in each following step, we construct not only the N th‐order inequality, but (formally) all the inequalities of order $M⩽N−1,$ so that these $(N+1)$ inequalities together form a set of alternating bounds $…⩽σ(−λ)k…$ These inequalities for $M⩽N−1$ actually turn out to be identical to those used in the inductive proof. See Sec. IIC.

Of course this choice must be done without breaking the ferromagnetic condition $Jxy⩾0$ (⁠$x≠y$⁠). This will be satisfied if (for example) we first set $Jxy = J>0$ for all $x≠y$ and $ax = a$ sufficiently large, and vary the $Cxy$ ’s in some bounded region including the initial values.

J.

Fröhlich

,

B.

Simon

, and

T.

Spencer

, , ().

J.

Fröhlich

,

R.

Israel

,

E. H.

Lieb

, and

B.

Simon

, , ().

J.

Fröhlich

,

R.

Israel

,

E. H.

Lieb

, and

B.

Simon

, , ().

J.

Rosen

, , ().

As for the present definition²⁶ of α. which reflects the behavior of full specific heat rather than its singular part, the inequality $α>0$ is valid for arbitrary systems in $Zd(d⩾3)$ with Griffiths‐class a priori measure. To prove the inequality, assume the converse, i.e., $C(J)→0$ as $J→Jc.$ Then from the Griffiths II inequalities, we have $〈φ0φ1〉crit = 〈φ02〉crit.$ (Hereafter $〈…〉crit$ denotes $limJ→Jc〈…〉J.$⁠) Inserting this into the Gell‐Mann‐Low representation³ $〈φ0φ(n,0,0,…)〉crit = (Φ0Ω,T″Φ0Ω)$ (T is a Hermitian operator with $|T|⩽1),$ we have $TΦ0Ω = Φ0Ω$ and thus $〈φ0φ(n,0,0,…)〉crit = 〈φ02〉crit$ for all n. This contradicts with the infrared bounds^3,21,22 $〈φ0φx〉crit⩽const |x|2−d.$ We are grateful to the referee for suggesting to us the present proof.

A. D.

Sokal

, , ().

Actually, Eq. (3.12) [and all the other similar relations such as Eq. (3.14)] must be proved in a finite lattice (with periodic boundary condition) for the corresponding quantities such as $χΛ≡Σx∈Λ〈φ0φx〉Λ.$ If one “works hard²” using some correlation inequalities (e.g., the Simon‐Lieb inequality^14,29,30 and the first‐order skeleton inequality), one can prove that these finite volume quantities converge to the infinite volume quantities (3.2)–(3.4) in the high‐temperature region (characterized by $χ<∞$⁠), and the relations carry over to the infinite systems.

E. H.

Lieb

, , ().

B.

Simon

, , ().

To prove this bound note that $J<a/(2d),$ a Gaussian system defined by letting $≶2n = 0(n⩾2)$ is well defined. Then the Griffiths II inequality implies $χ⩽χGussian<∞$ and the system is in its high‐temperature region.

J. L.

Lebowitz

, , ().

J.

Glimm

and

A.

Jaffe

, , ().