Magnetization in the zig-zag layered Ising model and orthogonal polynomials
aa r X i v : . [ m a t h - ph ] F e b MAGNETIZATION IN THE ZIG-ZAG LAYERED ISING MODELAND ORTHOGONAL POLYNOMIALS
DMITRY CHELKAK A , B , CL´EMENT HONGLER C , AND R´EMY MAHFOUF A Abstract.
We discuss the magnetization M m in the m -th column of the zig-zag layered 2D Ising model on a half-plane using Kadanoff–Ceva fermions andorthogonal polynomials techniques. Our main result gives an explicit repre-sentation of M m via m × m Hankel determinants constructed from the spectralmeasure of a certain Jacobi matrix which encodes the interaction parametersbetween the columns. We also illustrate our approach by giving short proofsof the classical Kaufman–Onsager–Yang and McCoy–Wu theorems in the ho-mogeneous setup and expressing M m as a Toeplitz+Hankel determinant forthe homogeneous sub-critical model in presence of a boundary magnetic field. Introduction
The planar Ising (or Lenz–Ising) model, introduced by Lenz almost a century ago,has an extremely rich history which is impossible to overview in a short introduction,instead we refer the interested reader to the monographs [40, 8, 45, 24] as wellas the papers [42, 44, 43, 36, 15] and references therein for more information onvarious facets of this history. From the ‘classical analysis’ viewpoint, one of theparticularly remarkable aspects is a very fruitful interplay between the explicitcomputations for the planar Ising model and the theory of Toeplitz determinants.This interplay originated in the groundbreaking work of Kaufman and Onsager inlate 1940s (see [6, 7]) and, in particular, lead Szeg¨o to the strong form of his famoustheorem on asymptotics of Toeplitz determinants; we refer the interested reader tothe recent survey [22] due to Deift, Its and Krasovsky for more information onthe developments of this link since then. It is nevertheless worth noting that thisresearch direction mostly originated in questions related to the homogeneous modelin the infinite-volume limit – a well-understood case from the physical perspective.At the same time, it seems that the much richer setup of the layered model – firstconsidered by McCoy–Wu and Au-Yang–McCoy in [38, 39, 34, 2, 3], see also [35,Sections 3.1,3.2] and [46] for historical comments – did not attract much attentionof mathematicians. Unfortunately, tour de force computations summarized in themonograph [40], are nowadays often considered (at least, in several mathematical
Mathematics Subject Classification.
Key words and phrases. planar Ising model, magnetization, discrete fermions, orthogonal poly-nomials, Hankel determinants, Toeplitz+Hankel determinants. A D´epartement de math´ematiques et applications, ´Ecole Normale Sup´erieure, CNRS,PSL University, 45 rue d’Ulm, 75005 Paris, France. B Holder of the ENS–MHI chair funded by MHI. On leave from St. Petersburg Dept.of Steklov Mathematical Institute RAS, Fontanka 27, 191023 St. Petersburg, Russia. C Chair of Statistical Field Theory, MATHAA Institute, ´Ecole PolytechniqueF´ed´erale de Lausanne, Station 8, 1015 Lausanne, Switzerland.
E-mail: [email protected] , [email protected] , [email protected] . sub-communities interested in 2D statistical mechanics) as being too technicallyinvolved to develop their analysis further. Certainly, this is an abnormal situationand by writing this paper we hope to bring the attention to this ‘layered’ setup,targeting not only probabilists but also the orthogonal polynomials community. Inthe mathematical physics literature, the interest to the layered Ising model alsoreappeared recently; e.g. see [1], [20] and references therein.Our paper should not be considered as a ‘39999th solution of the Ising model’.On the contrary, the methods we use can be viewed as a simplification of theclassical ones in presence of the translation and reflection symmetry in the directionorthogonal to the line connecting spins under consideration. Comparing to [40], thissimplification comes from the fact that we use the Kadanoff–Ceva lattice instead ofthe Onsager (or Fisher) one and, more importantly, work directly with orthogonalpolynomials instead of Toeplitz determinants. Though such details are not vital inthe homogeneous case, this allows us to perform computations for a general ‘zig-zag layered’ model in a transparent way (see Theorem 1.1); in the latter case, thepolynomials are orthogonal with respect to a certain measure on the segment [0 , real weights , the simplestpossible framework of the OPUC/OPRL theory. From the perspective of the ‘freefermion algebra’ solution [51] of the planar Ising model, our derivations can beviewed as its translation to the language of discrete fermionic observables, see [30]for a discussion of such a correspondence. The latter viewpoint was advertised bySmirnov in his celebrated work on the critical Ising model (e.g., see the lecturenotes [24] and references therein). We refer the interested reader to [15, Section 3]for a discussion of equivalences between various combinatorial formalisms used tostudy the planar Ising model, see also [41] and [18, Section 3.2]. In this paper wealso want to make a link between discrete complex analysis techniques and classicalcomputations more transparent.Before formulating our main result – Theorem 1.1 – for the layered Ising model,let us briefly mention the list of questions that we discuss along the way in thehomogeneous setup: • Kaufman–Onsager–Yang theorem on the spontaneous magnetization belowcriticality: Theorem 3.6, cf. [40, Section X.4]; • McCoy–Wu theorem on the asymptotic behavior of the horizontal spin-spincorrelations at criticality: Theorem 3.9, cf. [40, Section XI.5]; • the wetting phase transition in the subcritical model caused by a boundarymagnetic field [26, 50] (which was interpreted as a hysteresis effect in theearlier work [40]): we discuss a setup similar to [40, Section XIII] in Sec-tion 4.3 and reduce the problem to the analysis of explicit Toeplitz+Hankeldeterminants, see Theorem 4.4; • Wu’s explicit formula for diagonal spin-spin correlations in the fully homo-geneous critical Ising model (see [40, Section XI.4]) and magnetization inthe zig-zag half-plane: we provide a very short computation via Legendrepolynomials in the Appendix, note that we were unable to find neitherTheorem A.4 nor the identity (A.8) in the literature.We now move on to the layered Ising model in a half-plane. Instead of working inthe original framework of Au-Yang, McCoy and Wu, we slightly simplify the setup
IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 3 by considering the Ising model in the (left) half-plane on the π -rotated square gridwhich we call the zig-zag half-plane and denote by H ⋄ , see Fig. 4 for the notation.We believe that such a simplification does not change key features of the problem,at the same it allows us to obtain more transparent results in full generality. Weare mostly interested in making our main result – Theorem 1.1 – easily accessibleto the mathematical community interested in orthogonal polynomials rather thanin discussing the physics behind the problem. It is worth emphasizing that Theo-rem 1.1 does not express M m as a Toeplitz determinant. Nevertheless, we believethat the formula (1.5) is amenable for the asymptotic analysis and is of interestfrom the mathematical perspective.The (half-)infinite volume limit of the Ising model on H ⋄ is defined as a limitof probability measures on an increasing sequence of finite domains exhausting H ⋄ ,with ‘+’ boundary conditions at the right-most column C and at infinity. Allinteraction parameters between the columns C p − and C p are assumed to be thesame and equal to x p = exp[ − βJ p ] = tan θ p , where θ p ∈ (0 , π ) can be viewedas a convenient parametrization of βJ p , see Section 2.1 for more details. Let M m = M m ( θ , θ , . . . ) := E + H ⋄ [ σ ( − m − , ] (1.1)be the magnetization in the (2 m )-th column (the analysis for odd columns can bedone similarly). Denote D even := i cos θ cos θ . . . − sin θ sin θ cos θ cos θ . . . − sin θ sin θ cos θ cos θ . . .. . . . . . . . . . . . (1.2)and let D ∗ even = U even S even , S even =( D even D ∗ even ) / (1.3)be the polar decomposition of the operator D ∗ even , see also Remark 4.3 for anotherinterpretation of the (partial) isometry U even . Further, denote J := D even D ∗ even . Astraightforward computation shows that J = b − a . . . − a b − a . . . − a b . . .. . . . . . . . . . . . b k = cos θ k − cos θ k + sin θ k − sin θ k − ,a k = cos θ k − cos θ k sin θ k sin θ k +1 , (1.4)where θ := 0 and b = cos θ cos θ . Let ν J be the spectral measure of J associated with the first basis vector. It is easy to see that 0 ≤ J ≤ ν J ∈ [0 , µ on [0 , m [ µ ] := det[ R λ p + q µ ( dλ ) ] m − p,q =0 be the m -th Hankel determinant composed from the moments of this measure. Theorem 1.1.
For all θ , θ , . . . ∈ (0 , π ) and m ≥ , we have M m = | det P m U even P m | = det P m J / P m Q mk =1 cos θ k = H m [ λ / ν J ]( H m [ ν J ] · H m [ λν J ] ) / , (1.5) where U even is the (partial) isometry factor in the polar decomposition (1.3) , theJacobi matrix J = D even D ∗ even is given by (1.4) , and ν J is the spectral measure of J .Remark . Assume that θ k = θ for all k ≥
1, i.e., that we work with the fullyhomogeneous model. One can easily see thatsupp ν J = [cos (2 θ ) ,
1] if θ ≤ π while supp ν J = { } ∪ [cos (2 θ ) ,
1] if θ > π . DMITRY CHELKAK, CL´EMENT HONGLER, AND R´EMY MAHFOUF
In particular, this clearly marks the critical value θ crit = π of the interactionparameter. Moreover, in the supercritical regime θ > θ crit , the existence of anexponentially decaying eigenfunction ψ ◦ k = (cot θ ) k , ψ ◦ ∈ Ker D ∗ even , directly leadsto the exponential decay of the truncated determinants | det P m U even P m | . Remark . Assume now that θ k +2 n = θ k for all k ≥ n ≥
1. Inthis case, the criticality condition reads as Q nk =1 tan θ k = 1, see Lemma 5.2 below.This condition is equivalent to the fact that the continuous spectrum of J beginsat 0. Moreover, in this case the integrated density of states of the periodic Jacobimatrix J behaves like C J · π − √ λ as λ →
0, where C J = (cid:2) n − P nk =1 ( ψ ◦ k ) · P nk =1 ( a k ψ ◦ k ψ ◦ k +1 ) − (cid:3) / (1.6)and ψ ◦ k denotes the periodic vector solving the equation Jψ ◦ = 0. In Section 5.2we show that the quantity (1.6) also admits a clear geometric interpretation inthe context of the so-called s-embeddings of planar Ising models, see (5.8) and adiscussion following that identity.It is clear that the spectral properties of the matrix J (which can be viewed as aneffective propagator in the direction orthogonal to the boundary of H ⋄ ) are directlyrelated to the behavior of the magnetization M m as m → ∞ . Nevertheless, we arenot aware of asymptotical results for (1.5) in the general case, especially when J has a singular continuous spectrum . This leads to the following question: • to find necessary and sufficient conditions on the measure ν J that implythe asymptotics (a) lim inf m →∞ M m = 0 (b) lim sup m →∞ M m = 0 in (1.5).We believe that an answer to this question should shed more light, in particular,on the random layered 2D Ising model. Moreover, it would be very interesting • to understand the dynamics of the measure ν J when the inverse tempera-ture β varies from ∞ to 0 and hence all θ p = 2 arctan exp[ − βJ p ] increasefrom 0 to 1 in a coherent way.Classically, this dynamics should lead to the Griffits–McCoy phase transition fori.i.d. interaction constants J p and also could give rise to less known effects in thedependent case. As already mentioned above, one of the goals of this paper is tobring the attention of the probability and orthogonal polynomials communities tothese questions.The rest of the paper is organized as follows. In Section 2 we review theKadanoff–Ceva formalism of spin-disorder operators in the planar Ising model. InSection 3 we illustrate our approach by giving streamlined proofs of two classicalresults due to Kaufman–Onsager–Yang and McCoy–Wu, respectively: Theorem 3.6and Theorem 3.9. Though the proof of Theorem 1.1, presented in the forthcomingSection 4, is formally independent of Section 3, we believe that it should help thereader to position this proof into the classical Ising model landscape. In Section 5we briefly discuss the geometric interpretation of our results in the context of iso-radial embeddings of the critical Baxter’s Z-invariant model and its generalizations– s-embeddings of critical periodic Ising models – which were recently suggestedin [14]. Appendix is devoted to the explicit analysis of diagonal correlations (Wu’sformula) and the zig-zag half-plane magnetization at criticality via Legendre poly-nomials. IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 5
Acknowledgements.
We are grateful to Yvan Velenik for bringing our attentionto the papers [26, 50] on the wetting phase transition in the subcritical model,which was mentioned under the name hysteresis effect in the first version of our pa-per following the interpretation given in [40, Section XIII]. We also thank JacquesH.H. Perk for useful comments on the immense literature on the Ising model cor-relations. Several parts of this paper were known and reported since 2012/2013but caused a very limited interest, we are grateful to colleagues who encouraged usto carry this project out despite the circumstances. Dmitry Chelkak would like tothank Alexander Its, Igor Krasovsky, Leonid Parnovski and Alexander Pushnitskifor helpful discussions. The research of Dmitry Chelkak and R´emy Mahfouf waspartially supported by the ANR-18-CE40-0033 project DIMERS. Cl´ement Honglerwould like to acknowledge the support of the ERC SG CONSTAMIS, the NCCRSwissMAP, the Blavatnik Family Foundation and the Latsis Foundation.2.
Combinatorics of the planar Ising model
In order to keep the presentation self-contained, in this section we collect basicdefinitions and properties of the planar Ising model observables. Below we adoptthe notation from [14, 16], the interested reader is also referred to [15] or [29] formore details (note however that these papers use slightly different definitions). Eventhough we discuss the spin-disorder observables in the full generality ( m spins and n disorders), below we are interested in the situations m = n = 2 (Section 3 andAppendix) and m = 1, n = 2 (Section 4 and Appendix) only.2.1. Definition and domain wall representation.
Let G be a finite connected planar graph embedded into the plane such that all its edges are straight segments.We denote by G • the set of its vertices and by G ◦ the set of its faces (identifiedwith their centers). The (ferromagnetic) nearest-neighbor Lenz-Ising model on thegraph dual to G is a random assignment of spins σ u ∈ {± } to the faces u ∈ G ◦ such that the probability of a spin configuration σ = ( σ u ) is proportional to P G [ σ ] ∝ exp [ β P u ∼ w J e σ u σ w ] , e = ( uw ) ∗ , (2.1)where a positive parameter β = 1 /kT is called the inverse temperature , the sum istaken over all pairs of adjacent faces u, w (equivalently, edges e ) of G , and J = ( J e )is a collection of positive interaction constants , indexed by the edges of G . Belowwe use the following parametrization of J e : x e = tan θ e := exp[ − βJ e ] . (2.2)Note that the quantities x e ∈ (0 ,
1) and θ e := 2 arctan x e ∈ (0 , π ) have the samemonotonicity as the temperature β − .We let the spin σ out of the outermost face of G be fixed to +1, in other wordswe impose ‘ + ’ boundary conditions . In this case, the domain wall representation (also known as the low-temperature expansion ) of the Ising model is a 1-to-1 cor-respondence between spin configurations and even subgraphs P of G : given a spinconfiguration, P consists of all edges that separate pairs of disaligned spins. Onecan consider a decomposition (not unique in general) of P into a collection of non-intersecting and non-self-intersecting loops . The above correspondence implies that E G [ σ u . . . σ u m ] = Z − G P P ∈E G x ( P )( − loops [ u ,...,um ] ( P ) (2.3) DMITRY CHELKAK, CL´EMENT HONGLER, AND R´EMY MAHFOUF for u , . . . , u m ∈ G ◦ , where E G denotes the set of all even subgraphs of G , Z G := P P ∈E G x ( P ) , x ( P ) := Q e ∈ P x e , (2.4)and loops [ u ,...,u m ] ( P ) is the number (always well defined modulo 2) of loops in P surrounding an odd number of faces u , ..., u m . Up to a factor exp[ β P e ∈E G J e ], thequantity Z G is the partition function of the Ising model on G ◦ .2.2. Disorder insertions.
Following Kadanoff and Ceva [31], given an even num-ber of vertices v , . . . , v n ∈ G • we define the correlation of disorders µ v , . . . , µ v n h µ v . . . µ v n i G := Z − G · Z [ v ,...,v n ] G , Z [ v ,...,v n ] G := P P ∈E G ( v ,...,v n ) x ( P ) , (2.5)where E G ( v , ..., v n ) denotes the set of subgraphs P of G such that each of thevertices v , . . . , v n has an odd degree in P while all other vertices have an evendegree. Probabilistically, one can easily see that h µ v . . . µ v n i G = E G (cid:2) exp[ − β P ( uw ) ∗ ∈ P ( v ,...,v n ) J e σ u σ w ] (cid:3) , (2.6)where P ( v , . . . , v n ) is a fixed collection of edge-disjoint paths matching in pairsthe vertices v , . . . , v n ; note that the right-and side does not depend on the choiceof these paths. The Kramers–Wannier duality implies (e.g., see [31]) that h µ v . . . µ v n i G = E ⋆G • (cid:2) σ • v . . . σ • v n (cid:3) , (2.7)where the expectation in the right-hand side is taken with respect to the Isingmodel on vertices of G , with dual weights x e ∗ := tan ( π − θ e ) and free boundaryconditions. Indeed, (2.5) is nothing but the high-temperature expansion of (2.7).Similarly to Z G , one can interpret Z [ v ,...,v n ] G as the low-temperature (domainwalls) expansion of the partition function of the Ising model defined on the facesof a double cover G [ v ,...,v n ] of the graph G that branches over v , . . . , v n , with thefollowing spin-flip symmetry constraint : we require σ u σ u ⋆ = − u and u ⋆ lie over the same face in G . Using thisinterpretation, we introduce mixed correlations h µ v . . . µ v n σ u . . . σ u m i G := h µ v . . . µ v n i G · E G [ v ,...,vn ] [ σ u . . . σ u m ] , (2.8)where u , . . . , u m should be understood as faces of the double cover G [ v ,...,v n ] .Similarly to (2.6) one can easily give a probabilistic interpretation of these quantitiesin terms of the original Ising model on G . Nevertheless, we prefer to speak aboutthe Ising model on G [ v ,...,v n ] as this approach is more invariant and does not requireto fix an arbitrary choice of the disorder lines P ( v , . . . , v n ).By definition of the Ising model on G [ v ,...,v n ] , the correlation (2.8) fulfills thesign-flip symmetry constraint between the sheets of the double cover. When con-sidered as a function of both vertices v p and faces u q , this correlation is defined ona double cover of ( G • ) n × ( G ◦ ) m and changes the sign each time when one of thevertices v p ∈ G • turns around one of the vertices u q ∈ G ◦ (or vice versa). We call spinors functions defined on double covers that obey such a sign-flip property.2.3. Fermions and the propagation equation.
We need an additional notation.Let Λ( G ) := G • ∪ G ◦ be the planar bipartite graph whose set of faces ♦ ( G ) is in a1-to-1 correspondence with the set of edges of G . Let Υ( G ) denote the medial graphof Λ( G ), where the vertices of Υ( G ) are in a 1-to-1 correspondence with edges ( vu )of Λ( G ) and are also called corners of G , while the faces of Υ( G ) correspond eitherto vertices of G • or to vertices of G ◦ or to quads from ♦ ( G ). We denote by Υ × ( G ) a IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 7 double cover of the graph Υ( G ) that branches around each of its faces (e.g., see [41,Fig. 27] or [18, Fig. 6]). For a corner c = ( v ( c ) u ( c )) ∈ Υ × ( G ) (with u ( c ) ∈ G ◦ and v ( c ) ∈ G • ), let η c := i · exp[ − i arg( v ( c ) − u ( c ))] , (2.9)where the global prefactor i is chosen for later convenience. One can easily seethat η c is a spinor on Υ × ( G ) (which means that its values at the two points of Υ × ( G )lying over the same corner of G differ by a − Dirac spinor .Given c ∈ Υ × ( G ), one defines the Kadanoff–Ceva fermion as χ c := µ v ( c ) σ u ( c ) .More accurately, we set X ̟ ( c ) := h µ v ( c ) µ v . . . µ v n − σ u ( c ) σ u . . . σ u m − i G , (2.10)for ̟ := ( v , . . . , v n − , u , . . . , u m − ) ∈ ( G • ) n − × ( G ◦ ) m − . Let Υ × ̟ ( G ) denote adouble cover of Υ( G ) that branches over each of the faces of Υ( G ) except those cor-responding to the points from ̟ . The preceding discussion of mixed spin-disordercorrelations ensures that X ̟ is a spinor on Υ × ̟ ( G ). Finally, letΨ ̟ ( c ) := η c X ̟ ( c ) (2.11)where η c is defined by (2.9). The function Ψ ̟ locally does not branch (the signschanges of χ c and η c cancel each other). More precisely, Ψ ̟ is a spinor on thedouble cover of Υ ̟ ( G ) that branches only over points from ̟ : it changes the signonly when c turns around one of the vertices v p or the faces u q .We now move on to the crucial three-term equation for the correlations (2.10),called the propagation equation for Kadanoff–Ceva fermions on Υ × ( G ), see [47, 23,41] or [15, Section 3.5] for more details. For a quad z e ∈ ♦ ( G ) corresponding toan edge e of G , we denote its vertices by v ( z e ) ∈ G • , u ( z e ) ∈ G ◦ , v ( z e ) ∈ G • ,and u ( z e ) ∈ G ◦ , listed in the counterclockwise order. Further, for p, q ∈ { , } , let c p,q ( z e ) := ( v p ( z e ) u q ( z e )). The following identity holds for all triples of consecutive(on Υ × ̟ ( G )) corners c p, − q ( z e ), c p,q ( z e ) and c − p,q ( z e ) surrounding the edge e : X ̟ ( c p,q ) = X ̟ ( c p, − q ) cos θ e + X ̟ ( c − p,q ) sin θ e , (2.12)where θ e stands for the parametrization (2.2) of the Ising model weight x e of e . Inrecent papers, the equation (2.12) is often used in the context of rhombic lattices,in which case the parameter θ e admits a geometric interpretation (see Section 5.1),but in fact it does not rely upon a particular choice of an embedding (up to ahomotopy) of ♦ ( G ) into C provided that θ e is defined by (2.2).2.4. Cauchy–Riemann and Laplacian-type identities on the square grid.
From now on we assume that G is a subgraph of the regular square grid Z ⊂ C .In this situation one can use (2.12) to derive a version of discrete Cauchy–Riemannequations for the complex-valued observable Ψ ̟ defined by (2.11). Proposition 2.1.
Let c , d , c , d be corners of G located as in Fig. 1A (andlocated on the same sheet of the double cover Υ ̟ ( G ) ). Let θ , θ be the interactionparameters assigned via (2.2) to the edges e , e . Then, the following identity holds: [Ψ ̟ ( c ) cos θ − Ψ ̟ ( c ) sin θ ] = ± i · [Ψ ̟ ( d ) sin θ − Ψ ̟ ( d ) cos θ ] , (2.13) where the ‘ ± ’ sign is ‘ + ’ if the square ( c d c d ) is oriented counterclockwise (toppicture in Fig. 1A) and ‘ − ’ otherwise (bottom picture in Fig. 1A). DMITRY CHELKAK, CL´EMENT HONGLER, AND R´EMY MAHFOUF θ θ cd c d θθ c d cd (A) The notation used inProposition 2.1 (Cauchy–Riemann equations (2.13)). cc cc c + - ♭ (B) The notation used in Proposition 2.2 (mas-sive harmonicity of fermionic observables in thehomogeneous model away from the branchings). c o c ♭ o c + o c (v) + c - (v) c (v) - ♭ c (v) + ♭ c o - (C) The notation used in theproof of Lemma 3.2 (the value[∆ ( m ) X sym[ v , u ] ] near the point v ). c θθ θθ θ -- o + θ (cid:0)c(cid:1) - (cid:3)(cid:4) - (cid:5) (cid:6) (cid:7)(cid:8)(cid:9) (D) The notation used in Proposi-tion 2.4 (harmonicity-type identities inthe zig-zag layered model).
Figure 1.
Local relations for Kadanoff–Ceva fermionic observ-ables. We indicate the four ‘types’ of corners of (subgraphs of) thesquare grid by orienting the triangles depicting them accordingly.
Proof.
Let a be the center of the square ( c d c d ). Assume that c , d , c , d areneighbors of a on Υ × ̟ ( G ). Writing the two propagation equations at a one gets X ̟ ( c ) cos θ + X ̟ ( d ) sin θ = X ̟ ( a ) = X ̟ ( c ) sin θ + X ̟ ( d ) cos θ . Since η d = η d = e ± i π η a (with the same choice of the sign: ‘+’ for the left picture,‘ − ’ for the right one) and η c = η c = e ∓ i π η a , the result immediately follows. (cid:3) IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 9
Below we often focus on the values of observables Ψ ̟ or X ̟ at corners c ∈ Υ( G )of one of four ‘types’; by a type of a corner c we mean its geometric position insidethe face of G ⊂ Z to which c belongs, see 1. For a fixed type of corners, the values η c are all the same and, moreover, the branching structure of Υ × ̟ ( G ) restricted to thistype of corners coincides with the one of Υ ̟ ( G ). In other words, Ψ ̟ and X ̟ differonly by a global multiplicative constant on each of the four types of corners.In this paper, we are interested in the following two setups:– homogeneous model, in which all the parameters θ e corresponding to horizontaledges of Z have the common value θ h (resp., θ v for vertical edges);– zig-zag layered model on the π -rotated grid, in which all interaction constantsbetween each pair of adjacent columns have the same value (see Fig. 4).In both situations, one can use (2.13) to derive a harmonicity-type identity forthe values of X ̟ (note however that this is not possible in the general case). Proposition 2.2.
In the homogeneous setup, assume that a corner c ∈ Υ ̟ ( G ) isnot located near the branching, i.e., that neither v ( c ) nor u ( c ) are in ̟ . Then, theobservable X ̟ satisfies the following equation at c : X ̟ ( c ) = sin θ h cos θ v · [ X ̟ ( c + )+ X ̟ ( c − )] + cos θ h sin θ v · [ X ̟ ( c ♯ )+ X ̟ ( c ♭ )] , where c + , c ♯ , c − , c ♭ are the four nearby corners having the same type as c , locatedat the east, north, west and south direction from c , respectively (see Fig. 1B).Proof. Recall that, at corners of a given type, the values X ̟ and Ψ ̟ differ onlyby a multiplicative constant. Due to the symmetry of the homogeneous model, wecan assume that c, c + , c ♯ , c − , c ♭ are located as in Fig. 1B. One writes four Cauchy–Riemann equations (2.13) between c and c + , c and c ♯ , c and c − , c and c ♭ . Multi-plying the first equation by sin θ h , the second by cos θ h , the third by ( − cos θ v ) andthe fourth by ( − sin θ v ), and taking the sum gives the result. (cid:3) Remark . Proposition 2.2 can be reformulated as the massive harmonicity con-dition [∆ ( m ) X ̟ ]( c ) = 0, where the massive Laplacian ∆ ( m ) is defined as[∆ ( m ) F ]( c ) := − F ( c )+ sin θ h cos θ v · [ F ( c + )+ F ( c − )]+ cos θ h sin θ v · [ F ( c ♯ )+ F ( c ♭ )] . It is worth noting that ∆ ( m ) is a generator of a (continuous time) random walkon Z with killing rate 1 − sin( θ h + θ v ), thus one can easily guess the classicalcriticality condition θ h + θ v = π ⇔ sinh[2 βJ h ] · sinh[2 βJ v ] = 1basing upon Proposition 2.2. As demonstrated in [10], one can also use this massiveharmonicity property of fermionic observables X ̟ to derive the exponential rate ofdecay of spin-spin correlations above criticality.A similar identity holds in the layered setup (see Fig. 1D for the notation).Assume that c is a west corner of a face on the π -rotated square grid. Denote by c ♯ ± , c ♭ ± the four nearby corners of the same type as c and let θ − , θ ◦ and θ + be theparameters assigned via (2.2) to the edges to the left of c ⋆ − , to the left of c , and tothe left of c ⋆ + , respectively. Proposition 2.4.
In the setup described above (see also Fig. 1D), assume thatneither v ( c ) nor u ( c ) are in ̟ . Then, the following identity holds: X ̟ ( c ) = sin θ − cos θ ◦ · [ X ̟ ( c ♯ − )+ X ̟ ( c ♭ − )] + sin θ ◦ cos θ + · [ X ̟ ( c ♯ + )+ X ̟ ( c ♭ + )] . n v u D n + D n D n* + D n* - Figure 2.
To derive recurrence relations on horizontal spin-spincorrelations, we consider the Kadanoff–Ceva fermionic observ-able X [ v , u ] with two branchings at v = (0 , ) and u = ( n + , − ).The symmetrized observable X sym[ v , u ] is defined on north-west cor-ners (marked as ⊲ in the figure) and the anti-symmetrized observ-able X anti[ v , u ] is defined on north-east corners (marked as △ ). Proof.
The result follows by combining four Cauchy–Riemann equations (2.13) sim-ilarly to the proof of Proposition 2.2. (cid:3)
Remark . It is worth emphasizing that the harmonicity-type identities discussedin Propositions 2.2 and 2.4 fail when c is located near the branching. The reason isthat applying (2.13) four times one gets the difference X ̟ ( d ∗ ) − X ̟ ( d ) with d ∗ , d located over the same point on the different sheets of the double cover Υ ̟ ( G ).3. Homogeneous model
In this section we discuss classical results on the horizontal spin-spin correlationsin the infinite volume for the homogeneous model. Namely, we assume that allhorizontal edges have a weight exp[ − βJ h ] = tan θ h while all vertical edges havea weight exp[ − βJ v ] = tan θ v , see also Appendix in which the diagonal spin-spincorrelations are treated in the fully homogeneous critical case θ h = θ v = π . Thoughthese results and even a roadmap of the proofs are well-known (e.g., see the classicaltreatment by McCoy and Wu [40]), we use this setup to illustrate a simplificationthat comes from working directly with real-valued orthogonal polynomials insteadof Toeplitz determinants, an approach that we apply to the layered model.3.1. Full-plane observable with two branchings.
Assume that the square gridon which the Ising model lives is shifted so that its vertices coincide with Z × ( Z + )and the centers of faces are ( Z + ) × ( Z − ), see Fig. 2. It is well known (e.g.,see [25]) that there are no more than two extremal Gibbs measures (coming from ‘+’and ‘ − ’ boundary conditions at infinity) and that the spin correlations in the infinitevolume limit are translationally invariant. Given n ≥
0, we define the horizontal
IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 11 and next-to-horizontal correlations D n := E Z (cid:2) σ ( , − ) σ ( n + , − ) ] , D ⋆n := E ⋆ ( Z ) • [ σ • (0 , ) σ • ( n, ) ] ,D n +1 := E Z (cid:2) σ ( − , − ) σ ( n + , − ) ] , D ⋆n +1 := E ⋆ ( Z ) • [ σ • (0 , ) σ • ( n +1 , ) ] , e D n +1 := E Z [ σ ( − , ) σ ( n + , − ) ] , e D ⋆n +1 := E ⋆ ( Z ) • [ σ • (0 , ) σ • ( n +1 , − ) ] , where the expectations in the second column are taken for the dual Ising modelwith interaction parameters tan ( π − θ v ) and tan ( π − θ h ) assigned to horizontaland vertical edges of the dual square grid ( Z ) • , respectively. Due to (2.7) one canview these quantities as disorder-disorder correlations in the original model.Let v = (0 , ) and u = ( n + , − ). Below we rely upon the full-plane ob-servable X [ v , u ] which can be thought of as a (subsequential) limit of the similarobservables defined on finite graphs G exhausting the square grid. Indeed, since (cid:12)(cid:12) h µ v ( c ) µ v σ u ( c ) σ u i G (cid:12)(cid:12) ≤ h µ v ( c ) µ v i G = E ⋆G • [ σ • v ( c ) σ • v ] ≤ , (3.1)a point-wise subsequential limit exists; its uniqueness (and hence the existence ofthe true limit) follows from Lemma 3.1 given below. Moreover, in Section 3.2,we provide an explicit construction of functions satisfying the conditions listed inLemma 3.1, which allows us to identify X [ v , u ] with these explicit functions.Let [( Z ± ) × Z ; v , u ] denote the double cover of the lattice ( Z ± ) × Z branchingover v and u . We now introduce the following symmetrized and anti-symmetrized versions of the observable X [ v , u ] ( · ) on north-west and north-east corners, respec-tively (see Fig. 2): X sym[ v , u ] ( c ) := [ X [ v , u ] ( c ) + X [ v , u ] (¯ c )] , c ∈ [( Z + ) × Z ; v , u ] , (3.2) X anti[ v , u ] ( c ) := [ X [ v , u ] ( c ) − X [ v , u ] (¯ c )] , c ∈ [( Z − ) × Z ; v , u ] , (3.3)where the continuous conjugation z ¯ z on [( Z ± ) × Z ; v , u ] is defined so thatit maps the segment [ , n + ] × { } between v and u to itself (i.e., the conjugateof each point located over this segment is chosen to be on the same sheet of thedouble cover). Once z ¯ z is specified in between of the branching points, it can be‘continuously’ extended to the entire double cover [( Z ± ) × Z ; v , u ]. In particular,the points c located over the real line outside of the segment [ , n + ] are mappedby z z to their counterparts c ∗ on the other sheet of the double cover.We now list basic properties of the observables X sym[ v , u ] and X anti[ v , u ] and show thatthey characterize these observables uniquely. Due to (3.1) we have (cid:12)(cid:12) X sym[ v , u ] ( k + , s ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) X anti[ v , u ] ( k − , s ) (cid:12)(cid:12) ≤ k, s ∈ Z . Proposition 2.2 (see also Remark 2.3) ensures that the observables X sym[ v , u ] and X anti[ v , u ] are massive harmonic away from the branching points v , u . In particular, one has[∆ ( m ) X sym[ v , u ] ](( k + , s )) = 0 and [∆ ( m ) X anti[ v , u ] ](( k − , s )) = 0 if s = 0 . (3.4)Further, the spinor property of the observable X [ v , u ] together with the choice ofthe conjugation described above gives X sym[ v , u ] (( k + , , k [0 , n ] , [∆ ( m ) X sym[ v , u ] ](( k + , , k ∈ [1 , n − X anti[ v , u ] (( k − , , k ∈ [1 , n ] , [∆ ( m ) X anti[ v , u ] ](( k − , , k [0 , n +1] . (3.6) Finally, the definition of X [ v , u ] implies X sym[ v , u ] (( , D n , X sym[ v , u ] (( n + , D ⋆n ; (3.7) X anti[ v , u ] (( − , − D n +1 , X anti[ v , u ] (( n + , D ⋆n +1 , (3.8)where we assume that these pairs of corners are located on the same sheet of thedouble cover [( Z ± ) × Z ; v , u ] as viewed from the upper half-plane ; this is why thevalue D n +1 at ( − ,
0) appears with the different sign.
Lemma 3.1. (i) The uniformly bounded observable X sym[ v , u ] given by (3.2) is uniquelycharacterized by the properties (3.4) , (3.5) and its values (3.7) near v and u .(ii) Similarly, the uniformly bounded observable X anti[ v , u ] given by (3.3) is uniquelycharacterized by the properties (3.4) , (3.6) and its values (3.8) near v and u .Proof. (i) Let X and X be two bounded spinors satisfying (3.4),(3.5) and (3.7).Let ( Z k ) k ≥ be the random walk (with killing) started at c ∈ [( Z + ) × Z ; u , v ] thatcorresponds to the massive Laplacian ∆ ( m ) . This random walk almost surely hitsthe points located over the set { ( k + , , k [1 , n − }} or dies. Since the process( X − X )( Z k ) is a bounded martingale with respect to the canonical filtration, theoptional stopping theorem yields X ( c ) − X ( c ) = 0. The proof of (ii) is similar. (cid:3) The next lemma allows one to use an explicit construction of functions X [ v , u ] given in Section 3.2 in order to get a recurrence relation for the spin-spin correla-tions. For n ≥
1, denote L n := cos θ v · [ D n + cos θ h · e D n ] , L ⋆n := sin θ h · [ D ⋆n + sin θ v · e D ⋆n ] . (3.9) Lemma 3.2.
For each n ≥ , the following identities are fulfilled: − [∆ ( m ) X sym[ v , u ] ](( , L n +1 , − [∆ ( m ) X sym[ v , u ] ](( n + , L ⋆n +1 ; (3.10) − [∆ ( m ) X anti[ v , u ] ](( − , − L n , − [∆ ( m ) X anti[ v , u ] ](( n + , L ⋆n , (3.11) with the same choice of points on the double covers [( Z ± ) × Z ; v , u ] as above.If n = 0 , the identities (3.10) should be replaced by − [∆ ( m ) X sym[ v , u ] ](( , L + L ⋆ while (3.11) hold with L := cos θ v and L ⋆ := sin θ h .Proof. We focus on the first identity in (3.10). Let c ◦ = c ♭ + ( v ) := ( , X sym[ v , u ] ( c −◦ ) = 0 and hence − [∆ ( m ) X sym[ v , u ] ]( c ◦ ) = X [ v , u ] ( c ◦ ) − sin θ h cos θ v · X [ v , u ] ( c + ◦ ) − cos θ h sin θ v · [ X [ v , u ] ( c ♯ ◦ ) + X [ v , u ] ( c ♭ ◦ )] . Recall that we deduced the massive harmonicity property of the observables X [ v , u ] away from the branchings from four Cauchy–Riemann identities (2.13), each of thembased upon two propagation equations (2.12); see Fig. 1B. We now repeat the sameproof but with seven three-terms identities (2.12) instead of eight ones requiredto prove Proposition 2.2, the one involving the values of X [ v , u ] at c −◦ = ( − , c ♭ − ( v ) = ( − ,
0) and c ♯ − ( v ) = ( − , ) missing; see Fig. 1C. As a result, one seesthat the value [∆ ( m ) X sym[ v , u ] ]( c ♭ + ( v )) is cos θ v times the missing linear combinationof the values X [ v , u ] ( c ♭ − ( v )) = D n +1 and X [ v , u ] ( c ♯ − ( v )) · cos θ h = e D n +1 · cos θ h , IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 13 which leads to the first identity in (3.10) (we let the reader to check the signs ob-tained along the computation). The proofs of the other three identities for n ≥ n = 0, one should sum six three-term identities (2.12) when dealingwith X sym[ v , u ] and eight ones when dealing with X anti[ v , u ] . In the latter case, the val-ues L and L ⋆ appear due to the presence of the branchings v , u near the pointsat which ∆ ( m ) X anti[ v , u ] is computed (and the fact that D = D ⋆ = 1). (cid:3) Construction via the Fourier transform and orthogonal polynomials.
In this section we construct two bounded functions satisfying the properties (3.4)–(3.8) using Fourier transform and orthogonal polynomials techniques, the explicitformulas are given in Lemma 3.4 and Lemma 3.5. Recall that these explicit solutionsmust coincide with X sym[ v , u ] and X anti[ v , u ] due to Lemma 3.1. Instead of the doublecovers [( Z ± ) × Z ; u , v ], we work in the upper half-plane Z × N only (see Lemma 3.3for the link between the two setups).For a function V : Z × N → R we use the same definition of the massiveLaplacian [∆ ( m ) V ]( k, s ) as above for s ≥ N V ]( k,
0) := V ( k, − cos θ h sin θ v · V ( k, − sin θ h cos θ v · [ V ( k − ,
0) + V ( k +1 , V at the point ( k, P symn ] and [ P antin ] to solve. Due to Lemma 3.1,these problems are equivalent to constructing explicitly the functions X sym[ v , u ] and X anti[ v , u ] , respectively; see also Fig. 3. • [ P symn ] : given n ≥
1, to construct a bounded function V : Z × N → R such that the following conditions are fulfilled:[∆ ( m ) V ]( k, s ) = 0 if s ≥
1; [ N V ]( k,
0) = 0 for k ∈ [1 , n − V ( k,
0) = 0 for k [0 , n ]; V (0 ,
0) = D n and V ( n,
0) = D ⋆n . • [ P antin + ] : given n ≥
0, to construct a bounded function V : Z × N → R such that the following conditions are fulfilled:[∆ ( m ) V ]( k, s ) = 0 if s ≥
1; [ N V ]( k,
0) = 0 for k [0 , n +1]; V ( k,
0) = 0 for k ∈ [1 , n ]; V (0 ,
0) = − D n +1 ; V ( n +1 ,
0) = D ⋆n +1 . Lemma 3.3.
Assume that a function V sym n (resp., V anti n +1 ) solves the problem [P symn ] (resp., [P antin+1 ] ). Then, the following identities hold: [ N V sym n ](0 ,
0) = L n +1 , [ N V sym n ]( n,
0) = L ⋆n +1 ; (3.13)[ N V anti n +1 ](0 ,
0) = − L n , [ N V anti n +1 ]( n +1 ,
0) = L ⋆n . (3.14) Proof.
Consider a section of the double cover [( Z ± ) × Z ; v , u ] with a cut goingalong the horizontal axis outside the segment [0 , n + ] for the problem [P symn ] andalong [0 , n + ] for the problem [P antin+1 ]. Define two functions on north-west andnorth-east, respectively, corners of the grid by V sym[ v , u ] (( ± k + , s )) := V sym n ( k, s ) V anti[ v , u ] (( ± k − , s )) := ± V anti n +1 ( k, s ) . These functions vanish on the cuts and thus can be viewed as bounded spinors onthe double covers [( Z ± ) × Z ; v , u ], which satisfy all the conditions (3.4)–(3.8). The boundary valueproblem [P symn ] in theupper half-plane Z × N : n D n D n *N N N
00 0 0
The boundary valueproblem [ P antin+1 ] in theupper half-plane Z × N : NN D n+1 D n+1 n+1 - * Figure 3.
The symmetrized observable on north-west corners (seeFig. 2) and the anti-symmetrized observable on north-east onessolve the problems [ P symn ] and [ P antin+1 ], respectively, where thesign N denotes the Neumann boundary conditions [ N V ]( k,
0) = 0.Due to the uniqueness result provided by Lemma 3.1, this implies X sym[ v , u ] = V sym[ v , u ] and V anti[ v , u ] = V anti[ v , u ] . The identities (3.13), (3.14) now easily follow from (3.10),(3.11) and the definition (3.12). (cid:3) Let V be a solution to the problem [P symn ], recall that this solution is uniquedue to Lemma 3.1. To construct it explicitly, we start with a heuristic argument.Assume for a moment that the Fourier series b V s ( e it ) := P k ∈ Z V ( k, s ) e ikt , s ≥ , t ∈ [0 , π ] , are well-defined. The massive harmonicity property [∆ ( m ) V ]( k, s ) = 0 for s ≥ − sin θ h cos θ v cos t ] · b V s ( e it ) = cos θ h sin θ v · [ b V s − ( e it ) + b V s +1 ( e it )] . (3.15)A general solution to the recurrence relation (3.15) is a linear combination of thefunctions ( y − ( t ; θ h , θ v )) s and ( y + ( t ; θ h , θ v )) s , where 0 ≤ y − ≤ ≤ y + solve thequadratic equation[1 − sin θ h cos θ v cos t ] · y ( t ) = cos θ h sin θ v · [( y ( t )) + 1] . At level s = 0 we have b V ( e it ) = Q n ( e it ), an unknown trigonometric polynomialof degree n . Since we are looking for bounded Fourier coefficients of b V s , we aretempted to say that b V s ( e it ) = Q n ( e it ) · ( y − ( t ; θ h , θ v )) s for s ≥
1. A straightforwardcomputation shows that P k ∈ Z [ N V ]( k, e ikt = w ( t ; θ h , θ v ) Q n ( e it ) , (3.16) w ( t ; θ h , θ v ) := (cid:2) (1 − sin θ h cos θ v cos t ) − (cos θ h sin θ v ) (cid:3) / . (3.17)The key observation of this section is that the left-hand side of (3.16) should notcontain monomials e it , . . . , e i ( n − t , which is a simple orthogonality condition forthe polynomial Q n ( e it ).We now use the heuristics developed in the previous paragraph to rigorouslyidentify the unique solution to [P symn ]. Lemma 3.4.
Let n ≥ . If a trigonometric polynomial Q n ( e it ) = D n + . . . + D ⋆n e int of degree n with prescribed free and leading coefficients is orthogonal to the family IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 15 { e it , . . . , e i ( n − t } with respect to the measure w ( t ; θ h , θ v ) dt π , then the function V ( k, s ) := π R π − π e − ikt Q n ( e it )( y − ( t ; θ h , θ v )) s dt is uniformly bounded and solves the problem [P symn ] . Moreover, h Q n , i w π dt = L n +1 and h Q n , e int i w π dt = L ⋆n +1 , (3.18) where the scalar product is taken with respect to the same measure on the unit circle.Proof. The values V ( k, s ) are uniformly bounded as 0 ≤ y − ≤
1, the massive har-monicity property [∆ ( m ) V ]( k, s ) = 0 for s ≥ V ( k,
0) and [ N F ]( k,
0) follow from the assumptions madeon the polynomial Q n . The identities (3.13) give (3.18). (cid:3) A similar construction can be done for the problem [P antin+1 ], see Fig. 3. Theonly difference is that at level s = 0 we now require that b V ( e it ) does not containmonomials e it , . . . , e i ( n +1) t while P k ∈ Z [ N V ]( k, e ikt = w ( t ; θ h , θ v ) b V ( e it ) = − L n + . . . + L ⋆n e i ( n +1) t (3.19)is a trigonometric polynomial of degree n + 1. In other words, this polynomial isorthogonal to { e it , . . . , e int } with respect to the weight w ( t ; θ h , θ v ) := ( w ( t ; θ h , θ v )) − , t ∈ [0 , π ] . (3.20)provided that w is integrable on the unit circle. One can easily see from (3.17)that this is true if and only if θ h + θ v = π . We discuss a modification of the nextclaim required for the analysis of the critical case θ h + θ v = π in Section 3.4. Lemma 3.5.
Let n ≥ and assume that θ h + θ v = π . If a trigonometric polynomial Q n +1 ( e it ) = − L n + . . . + L ⋆n e i ( n +1) t is orthogonal to the family { e it , . . . , e int } withrespect to the measure w ( t ; θ h , θ v ) dt π , then the function V ( k, s ) := π R π − π e − ikt Q n +1 ( e it )( y − ( t ; θ h , θ v )) s w ( t ; θ h , θ v ) dt (3.21) is uniformly bounded and solves the problem [P antin+1 ] . Moreover, h Q n +1 , i w π dt = − D n +1 and h Q n +1 , e i ( n +1) t i w π dt = D ⋆n +1 , (3.22) where the scalar product is taken with respect to the same measure on the unit circle.Proof. The proof repeats the arguments used in the proof of Lemma 3.4. (cid:3)
Horizontal spin-spin correlations below criticality.
In this section wecombine the results of Lemmas 3.4 and 3.5 into a single result on asymptotics ofthe horizontal spin-spin correlations D n as n → ∞ . We assume that θ h + θ v < π and rely upon the fact that D ⋆n → n → ∞ . This can be easily derived from themonotonicity of D n with respect to the temperature and the fact that D n = D ⋆n → n → ∞ in the critical regime θ h + θ v = π which is discussed in the next section. Theorem 3.6 ( Kauffman–Onsager–Yang).
Let θ h + θ v < π . Then, the spon-taneous magnetization M ( θ h , θ v ) of the homogeneous Ising model is given by M ( θ h , θ v ) := lim n →∞ D / n = (cid:2) − (tan θ h tan θ v ) (cid:3) / . (3.23) (Note that under the parametrization (2.2) one has tan θ e = (sinh(2 βJ e )) − .) Remark . It is worth mentioning that the value tan θ h tan θ v also admits afully geometric interpretation as Baxter’s elliptic parameter of the Z-invariant Isingmodel on isoradial graphs [8, Eq. (7.10.50)], see Section 5.1 for details. Proof.
Classically, the computation given below is based upon the strong Szeg¨otheorem on the asymptotics of the norms of orthogonal polynomials on the unitcircle. Note however that we use this result in its simplest form, for real weights w and w given by (3.17) and (3.20).Let Φ n ( z ) = z n + . . . − α n − be the n -th monic orthogonal polynomial on theunit circle with respect to the measure w ( t ; θ h , θ v ) dt π , the real number α n − iscalled the Verblunsky coefficient , recall that | α n − | < n ≥
1. Denoteby Φ ∗ n := z n Φ n ( z − ) = − α n − z n + . . . + 1 the reciprocal polynomial. Matchingthe free and the leading coefficients, it is easy to see that the polynomial Q n fromLemma 3.4 can be written as Q n ( e it ) = c n Φ n ( e it )+ c ∗ n Φ ∗ n ( e it ) , where (cid:20) c ∗ n c n (cid:21) = (cid:20) − α n − − α n − (cid:21) − (cid:20) D ⋆n D n (cid:21) . Moreover, one has h Φ n , e int i = h Φ ∗ n , i = k Φ n k =: β n = β Q nk =1 (1 − α k − )(e.g., see [52, Theorem 2.1]) and h Φ n , i = h Φ ∗ n , e int i = 0, here and below we dropthe measure w dt π from the notation for shortness. Therefore, the identities (3.13)imply that (cid:20) L ⋆n +1 L n +1 (cid:21) = β n (cid:20) c ∗ n c n (cid:21) = β n − (cid:20) α n − α n − (cid:21) (cid:20) D ⋆n D n (cid:21) , (3.24)and hence L n +1 − ( L ⋆n +1 ) = β n β n − · ( D n − ( D ⋆n ) ) for n ≥ . (3.25)Similarly, it follows from Lemma 3.5 that (cid:20) D ⋆n +1 − D n +1 (cid:21) = β n (cid:20) α n α n (cid:21) (cid:20) L ⋆n − L n (cid:21) , (3.26)where α n and β n stand for the Verblunsky coefficients and squared norms of monicorthogonal polynomials corresponding to the weight (3.20). In particular, we have D n +1 − ( D ⋆n +1 ) = β n +1 β n · ( L n − ( L ⋆n ) ) for n ≥ . (3.27)The recurrence relations (3.27), (3.25) applied for even and odd indices n , respec-tively, lead to the formula D m +1 − ( D ⋆ m +1 ) = β m +1 β m · β m − β m − · ( D m − − ( D ⋆ m − ) )= . . . = Q m +1 k =0 β k · Q m − k =0 β k · ( L − ( L ∗ ) ) , note that L − ( L ∗ ) = (cos θ v ) − (sin θ h ) = cos( θ h + θ v ) cos( θ h − θ v ).Recall that D ⋆ m +1 → m → ∞ . It remains to apply the Szeg¨o theory (e.g.,see [28, Section 5.5] or [52, Theorems 8.1 and 8.5]) to the weights (3.17) and (3.20).A straightforward computation shows that w ( t ; θ h , θ v ) = Cw q − ( t ) w q + ( t ) , where C = (cos θ h ) cos θ v ,w q ( t ) := (cid:2) (1+ q ) − (2 q cos t ) (cid:3) / = | − q e it | , q ± = tan( θ h ) tan( π ∓ θ v ) . Since w ( t ; θ h , θ v ) = ( w ( t ; θ h , θ v )) − , we havelim m →∞ Q m +1 k =0 β k · Q m − k =0 β k = C − · G , IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 17 where G = exp h π Z Z D (cid:12)(cid:12)(cid:12) ddz (cid:0) log(1 − q − z ) + log(1 − q z ) (cid:12)(cid:12)(cid:12) dA ( z ) i = exp (cid:2) − P k ≥ k ( q k − + q k + ) (cid:3) = (cid:2) (1 − q − )(1 − q )(1 − q − q ) (cid:3) − / = (cos θ h ) (cos θ v ) / (cos θ h ) − / (cos( θ h + θ v ) cos( θ h − θ v )) − / . Putting all the factors together, one gets (3.23). (cid:3)
Remark . The identity (3.26) with n = 0 also provides a formula D = β · [cos θ v − α sin θ h ]for the energy density (on a vertical edge) of the homogeneous Ising model.3.4. Asymptotics of horizontal correlations D n as n → ∞ at criticality. Assume now that θ h + θ v = π . Another classical result that we discuss in thissection is that spin-spin correlations D m decay like m − / at large distances. Theorem 3.9 ( McCoy–Wu).
Let C σ := 2 e ζ ′ ( − , θ h = θ and θ v = π − θ .Then, D m ∼ C σ · (2 m cos θ ) − / as m → ∞ . (3.28) Proof.
A straightforward computation shows that w ( t ; θ, π − θ ) = 2 sin θ · [1 − (sin θ cos t ) ] / · | sin t | . In particular, the weight w := w − is not integrable and the arguments used in theproof of Theorem 3.6 require a modification. Also, the Kramers–Wannier dualityensures that D n = D ⋆n , L n = L ⋆n and hence the identities (3.25), (3.27) becomeuseless (though one could still could use (3.24)). In this situation we prefer toswitch to the framework of orthogonal polynomials on the real line (more precisely,on the segment [ − , w ( x ; θ ) := [ 1 − (sin θ · x ) ] / , x ∈ [ −
1; 1] , (3.29)and let P n ( x ) = x n + . . . be the monic orthogonal polynomial of degree n on [ − , w ( x, θ ). It is easy to check that the trigonometricpolynomial Q n ( e it ) := D n · e int · n P n (cos t )fits the construction given in Lemma 3.4 to solve the problem [P symn ]. The for-mula (3.18) gives L n +1 = 12 π Z π − π Q n ( e it ) w ( t ; θ, π − θ ) dt = D n n − π Z π − π cos( nt ) P n (cos t ) w ( t ; θ, π − θ ) dt = D n n +1 sin θπ Z − (2 n − x n + . . . ) P n ( x ) w ( x ; θ ) dx = π − n sin θ · k P n k wdx · D n , n ≥ . (3.30)Moreover, a similar computation for n = 0 implies that2 L = 2 π − sin θ R − P ( x ) w ( x ; θ ) dx = 2 π − sin θ · k P k wdx (3.31)since D = 1 and due to the modification required in Lemma 3.2 in the case n = 0. We can use the same line of reasoning to construct a solution of the prob-lem [P antin+1 ] treated in Lemma 3.5 in the non-critical regime. Namely, let P n ( x )be the monic orthogonal polynomial of degree n on [ − ,
1] with respect to theweight w ( x ) := [ 1 − (sin θ · x ) ] − / , x ∈ [ − , , (3.32)and Q n +1 ( e it ) := L n · ( e it − e int · n P n (cos t ) . It is straightforward to check that the formula (3.21) gives a solution to the bound-ary value problem [P antin+1 ], note that the product ( e it − w ( t ; θ, π − θ ) is integrableon the unit circle as the first factor kills the singularity of w at t = 0. Moreover,the computation (3.22) remains valid and reads as D n +1 = − π Z π − π Q n +1 ( e it ) w ( t ; θ, π − θ ) − dt = L n n π Z π − π sin( ( n +1) t )sin t P n (cos t ) (sin t ) dtw ( t ; θ, π − θ )= L n n π sin θ Z − (2 n x n + . . . ) P n ( x ) w ( x ) dx = π − n (sin θ ) − · k P n k w dx · L n , n ≥ . (3.33)Recall that L = sin θ (see Lemma 3.2). Taking a product of the recurrencerelations (3.31), (3.30) for n = 1 , . . . , m −
1, and (3.33) for n = 0 , . . . , m , one obtainsthe identity D m +1 D m = π − m − m Y m − k =0 k P k k wdx · Y mk =0 k P k k w dx , (3.34)where the weights w ( x ; θ ) and w ( x ; θ ) on [ − ,
1] are given by (3.29) and (3.32).This is again a classical setup of the orthogonal polynomials theory, note that ifone now passes back to the unit circle than the | t | -type singularity of the weightsappear at the point e it = 1. One might now use the general results (accounted, e.g.,in [21]) but we prefer to refer to a specific treatment [5]. Applying [5, Theorem 1.7]with parameters α = 0 , β = ± and k = sin θ one obtains the asymptotics D m +1 D m ∼ π [ G ( )] (1 − k ) − / m − / ∼ / e ζ ′ ( − (2 m cos θ ) − / , m → ∞ , where G denotes the Barnes G-function. (Note that [5] also provides sub-leadingterms of this asymptotics.) The proof of (3.28) is complete modulo the factthat D m +1 ∼ D m as m → ∞ . This statement can be proved by the argumentsgiven in the next remark (or, alternatively, using probabilistic estimates). (cid:3) Remark . Due to the famous quadratic identities [47, 37] for the spin-spincorrelations (recall also the definition (3.9) of L n ), one can write (3.30) and (3.33)as A n := π − n k P n k wdx = D n +1 +cos θ · e D n +1 D n = D n +2 D n +1 − cos θ · e D n +1 ,B n +1 := π − n +2 k P n +1 k w dx = D n +2 D n +1 +cos θ · e D n +1 = D n +1 − cos θ · e D n +1 D n . IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 19
In fact, one can also prove these identities by considering the anti-symmetrization(resp., symmetrization) of the observable X [ u , v ] on the north-west (resp., north-east) corners of the lattice and noticing that, up to a multiplicative constant, itsolves the problem [P antin+2 ] (resp., [P symn − ]). In particular, we have D m +1 /D m = ( A m + B m +1 ) = 2( A − m − + B − m ) − so one can see that D m +1 ∼ D m and find sub-leading corrections to the asymptoticsof D m (and e D m ) using the analysis of orthogonal polynomials performed in [5].4. Layered model in the zig-zag half plane
In this section we work with the (half-)infinite volume limit of the Ising modelon the zig-zag half-plane H ⋄ (see Fig. 4 for the notation), which is defined as a limitof probability measures on an increasing sequence of finite domains exhausting H ⋄ ,with ‘+’ boundary conditions at the right-most column C and at infinity. Allinteraction parameters between the columns C p − and C p are assumed to be thesame and equal to x p = exp[ − βJ p ] = tan θ p . The goal is to find a representationfor the magnetization M m at the column C m , see (1.1). The uniqueness of therelevant half-plane fermionic observable is discussed in Section 4.1 and our mainresult – Theorem 1.1 – is proved in Section 4.2. In Section 4.3 we use Theorem 1.1to discuss the wetting phase transition [26, 50] caused by a boundary magnetic field.In this case the Jacobi matrix J can be explicitly diagonalized and the final answercan be written in terms of the so-called Toeplitz+Hankel determinants.4.1. Half-plane fermionic observable.
Let v = ( − m − , X [ v ] defined by (2.10); comparing with Section 3 onecan think about the spin σ u := σ out as being attached to the vertical boundary.Below we are mostly interested in the values of X [ v ] at west corners (see Fig. 4) H ( − k, s ) := Ψ [ v ] (( − k, s )) = X [ v ] (( − k, s )) , k ∈ N , s ∈ Z , k + s Z , note the convention on η c chosen in (2.9). By definition, one has H ( − m − ,
0) = E + H ⋄ [ σ ( − m − , ] = M m . (4.1)We also need the values of X [ v ] at east corners: H ◦ ( − k, s ) := Ψ [ v ] (( − k, s )) = iX [ v ] (( − k, s )) , k ∈ N , s ∈ Z , k + s ∈ Z . It is convenient to set θ := 0 and H ◦ (0 , s ) := 0 for all s ∈ Z .The infinite-volume observable X [ v ] is defined as a (subsequential) limit of thesame observables constructed in finite regions. Subsequential limits exist due tothe uniform bound (3.1) while the uniqueness of X [ v ] is given by Lemma 4.1. Thediscrete Cauchy–Riemann identities (2.13) can be written as H ( − k − , s ±
1) sin θ k +1 − H ( k, s ) cos θ k = ± i · [ H ◦ ( − k, s ±
1) sin θ k − H ◦ ( − k − , s ) cos θ k +1 , k ≥ , k + s Z . (4.2)Near the vertical boundary, these equations should be modified as follows: H ( − , s ±
1) sin θ − H (0 , s ) = ∓ i · H ◦ ( − , s ) cos θ , s Z . (4.3)Indeed, X [ v ] (( − , s ± )) = X [ v ] (0 , s )) = H (0 , s ) and hence (4.3) are nothing butthe three-term identities (2.12). θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ v M m iM m + CCCCCCCC
Figure 4.
The zig-zag layered model in the left half-plane H ⋄ .All the interaction parameters between two adjacent columns areassumed to be the same. The ‘+’ boundary conditions are imposedat the column C . To analyze the ratio M m +1 /M m we consider theKadanoff–Ceva fermionic observable branching at v = ( − m − , Lemma 4.1.
The spinors
H, H ◦ defined in H ⋄ and branching over v are uniquelydetermined by the following conditions: uniform boundedness, Cauchy–Riemannidentities (4.2) , boundary relations (4.3) , and the value (4.1) of H near v .Proof. Taking the difference of two solutions, assume that
H, H ◦ are uniformlybounded, satisfy (4.2), (4.3) and that H ( − m − ,
0) = 0. Recall that Proposi-tion 2.4 gives the harmonicity-type identity H ( − k, s ) = sin θ k +1 cos θ k · [ H ( − k − , s +1) + H ( − k − , s − sin θ k cos θ k − · [ H ( − k +1 , s +1) + H ( − k +1 , s − c = ( − k + , s ) with k ≥ k = − m − s = 0 (i.e., at the west corner located near the branching v ). Moreover, due to theboundary relations (4.3), exactly the same identity holds for k = 0 , θ := 0). In its turn, the function H ◦ satisfies the identities H ◦ ( − k, s ) = cos θ k +1 sin θ k · [ H ◦ ( − k − , s +1) + H ◦ ( − k − , s − cos θ k sin θ k − · [ H ◦ ( − k +1 , s +1) + H ◦ ( − k +1 , s − IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 21 k k-1k+1 cos (cid:10) k ( ) sin (cid:11) k ( ) cos (cid:12) k ( ) sin (cid:13) k ( ) CCCCC
CCCCC
Figure 5.
The functions ̺ k H ( − k, s ) and ̺ ◦ k H ◦ ( − k, s ) are discreteharmonic with respect to the random walks having these transitionprobabilities in the horizontal direction (and in the vertical one).at all east corners d = ( − k + , s ), including the one located near the branching v (in the latter case the proof of Proposition 2.4 works verbatim due to the factthat H ( − m − ,
0) = 0). Both (4.4) and (4.5) can be rewritten as true discreteharmonicity properties if one passes from H and H ◦ to the functions e H ( − k, s ) := ̺ k · H ( − k, s ) , e H ◦ := ̺ ◦ k · H ◦ ( − k, s ) ,̺ k := Q kj =1 (sin θ j / cos θ j − ) , ̺ ◦ k := Q kj =2 (cos θ j / sin θ j − ) , recall that we set e H ◦ (0 , s ) = H ◦ (0 , s ) := 0 on the vertical axes.Let Z n = ( K n , S n ) (resp., Z ◦ n = ( K ◦ n , S n )) be the nearest-neighbor random walkon west (resp., east) corners, with jump probabilities ( , ) for the process S n and(cos θ k , sin θ k ) for the process K n (resp., (sin θ k , cos θ k ) for the process K ◦ n ), seeFig. 5. Note that the walk Z n on west corners is reflected from the vertical axeswhile the walk Z ◦ n on east corners is absorbed there.It follows from (4.4) that the stochastic process e H ( Z n ) is a martingale, whenequipped with the canonical filtration, until the first time when Z n hits the westcorner ( − m − ,
0) located near the branching, recall that e H ( − m − ,
0) = 0.Similarly, (4.5) implies that the process e H ◦ ( Z ◦ n ) is a martingale until the first timewhen Z ◦ n hits the imaginary axis, recall that e H ◦ = 0 there. As we show below,depending on the behavior of ̺ k and ̺ ◦ k as k → ∞ , the optional stopping theoremallows to conclude that either e H or e H ◦ vanishes identically. Once the identity e H ≡ e H ◦ ≡
0) is proven, the equations (4.2), (4.3) and the fact that e H ◦ vanishes onthe imaginary axis (resp., e H vanishes at the point ( − m − , e H ◦ ≡ e H ≡
0) too. Recall that the functions H and H ◦ are uniformly boundedand note that ̺ k ̺ ◦ k = (cos θ ) − sin θ k cos θ k = O (1) as k → ∞ . It follows from themaximum principle that • the function e H is uniformly bounded unless ̺ k → ∞ as k → ∞ ; • the function e H ◦ is uniformly bounded unless ̺ ◦ k → ∞ as k → ∞ .We have three cases to consider separately. • Let lim inf k →∞ ̺ k = 0, in particular this implies that e H is uniformlybounded. The optional stopping theorem applied to the martingale e H ( Z n )and the fact that a one-dimensional random walk on − N reflected at 0almost surely takes arbitrary large (negative) values imply that H ≡ • Let lim inf k →∞ ̺ ◦ k = 0. A similar argument applied to the martingale e H ◦ ( Z ◦ n )(recall that e H ◦ vanishes on the imaginary axis) shows that e H ◦ ≡ • Let both sequences ̺ k and ̺ ◦ k be uniformly bounded from below as k → ∞ .Since ̺ k ̺ ◦ k = (cos θ ) − sin θ k cos θ k , these sequences are also uniformlybounded from above and the parameters θ k , k ≥
1, stay away from 0. In this case it is easy to see that the process K ◦ n hits 0 almost surely (i.e., thatthe random walk Z ◦ n hits the imaginary axis almost surely). Indeed, theprobability p ◦ k to hit 0 starting from − k satisfies the recurrence p ◦ k − p ◦ k +1 = cot θ k · ( p ◦ k − − p ◦ k ) = . . . = ̺ − k +1 sin θ k +1 · (1 − p ◦ ) , which is only possible if p ◦ = 1 since the factors ̺ k +1 / sin θ k +1 are uniformlybounded. We conclude as before by applying the optional stopping theoremto the uniformly bounded martingale e H ( Z ◦ n ).The proof is complete. (cid:3) Magnetization M m in the (2 m )-th column. Similarly to Section 3.2,below we rely upon the uniqueness Lemma 4.1 and aim to construct the valuesof X [ v ] on west and east corners (i.e., the pair of spinors H, H ◦ ) as explicitly aspossible. Note that we have H ( − p − ,
0) = 0 for p ≥ m +1 , H ◦ ( − p,
0) = 0 for p ≤ m. (4.6)since the spinors defined (on the double cover branching over v ) by the symmetry H ( − k, − s ) := H ( − k, s ), H ◦ ( − k, − s ) := − H ◦ ( − k, s ) also satisfy the Cauchy–Riemann equations (4.2), (4.3) and thus must coincide with H, H ◦ .Given s ≥
0, let H s denote the semi-infinite vector of the (real) values H ( − k, s ), k ∈ N , where we assign zero values to the indices s such that s + k ∈ Z . Similarly,let H ◦ s be the vector of the (purely imaginary) values H ◦ ( − k, s ), k ∈ N , wherewe assign zero values to the indices s such that s + k Z . We can write theharmonicity-type equations (4.4) and (4.5) as H s = C [ H s − + H s +1 ] , H ◦ s = C ◦ [ H ◦ s − + H ◦ s +1 ] , s ≥ , (4.7)where the self-adjoint operators C and C ◦ are given by C := θ . . . sin θ θ cos θ . . . θ cos θ θ cos θ . . . θ cos θ . . .. . . . . . . . . . . . . . . ,C ◦ := θ sin θ . . . cos θ sin θ θ sin θ . . . θ sin θ θ sin θ . . . θ sin θ . . .. . . . . . . . . . . . . . . . Let T ( λ ) := λ − · (1 − √ − λ ). Similarly to Section 3.2, in order to satisfy therecurrences (4.7) we intend to write H s := [ T ( C )] s H , H ◦ s := [ T ( C )] s H ◦ , s ≥ . (4.8)We now introduce an operator D , which plays the key role in the rest of the analysis: D := i cos θ . . . θ cos θ . . . − sin θ sin θ θ cos θ . . . − sin θ sin θ θ cos θ . . .. . . . . . . . . . . . . . . . IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 23
A straightforward computation gives CD = DC ◦ , DD ∗ = I − C and D ∗ D = I − ( C ◦ ) . (4.9)In particular, this implies that − I ≤ C, C ◦ ≤ I . Thereofer, the operators T ( C )and T ( C ◦ ) in (4.8) are well-defined and the vectors H s and H ◦ s defined by (4.8)are uniformly bounded as s → ∞ . Still, we need to find the vectors H and H ◦ so that not only the harmonicity-type identities (4.7) for H and H ◦ but also theCauchy–Riemann equations (4.2), (4.3) relating H s and H ◦ s are satisfied.Note that Ker D = { } while the kernel of D ∗ might be two-dimensional.Let D ∗ = U ( DD ∗ ) / be the polar decomposition of D ∗ , where U := ( D ∗ D ) − / D ∗ = D ∗ ( DD ∗ ) − / (4.10)is a (partial) isometry . We are now able to formulate the key proposition on theconstruction of solutions to (4.2), (4.3) in the upper quadrant. Proposition 4.2.
Given H ∈ ℓ , let H ◦ := U H . Then, H s := [ T ( C )] s H and H ◦ s := [ T ( C ◦ )] s H ◦ are uniformly bounded in ℓ and provide a solution tothe Cauchy–Riemann equations (4.2) , (4.3) in the upper quadrant.Proof. Since − I ≤ C, C ◦ ≤ I , we have 0 ≤ T ( C ) , T ( C ◦ ) ≤ I . Therefore, H s and H ◦ s are uniformly bounded in ℓ . Moreover, (4.9) and (4.10) imply that U C = C ◦ U and hence H ◦ s = [ T ( C ◦ )] s U H = U [ T ( C )] s H = U H s for all s ≥
0. This allows oneto write CH s +1 − H s = − ( I − C ) / H s = − DU H s = − DH ◦ s , (4.11) H s +1 − CH s = − ( I − C ) / H s +1 = − DH ◦ s +1 . (4.12)It is not hard to see that these equations are equivalent to the Cauchy–Riemannidentities (4.2), (4.3). Indeed, the first entry of the vector-valued equation (4.11)or (4.12) (depending on the parity of s ) gives the relation (4.3) while the first entryof the other equation gives a linear combination of (4.3) and (4.2) with k = 1.Further, each of the next entries of (4.11) and (4.12) gives a linear combination oftwo identities (4.2) with two consecutive k ’s. Therefore, for each s ≥ k ) recover all the identities (4.3), (4.2) from (4.11) and (4.12). (cid:3) Clearly, the operators D and U can be split into independent components in-dexed by odd/even indices, only one of which is relevant for the value of the mag-netization M m in the even columns C m , the other component is responsible forthe magnetization in odd columns. In particular, the relevant block D even of theoperator D is given by (1.2). Remark . In view of the result provided by Proposition 4.2, the (partial) isometry U even can be thought of as a discrete Hilbert transform associated with the Cauchy–Riemann equations (4.2), (4.3) in the upper quadrant: given the values H of thereal part of a ‘discrete holomorphic’ function ( H, H ◦ ) on the real line, it returnsthe boundary values H ◦ = U even H of its imaginary part.We are now able to prove the main result of this section. Proof of Theorem 1.1.
Let H and H ◦ be the values of the half-plane observable X [ v ] on west and east corners, respectively. Since H is a finite vector (see (4.6)), it belongs to ℓ . Therefore, Lemma 4.1 and Proposition 4.2 imply that DH ◦ = DU H = ( DD ∗ ) / · [ ∗ . . . ∗ M m . . . ] ⊤ , where we use the symbol ∗ to denote unknown entries of H . On the other hand,note that − iH ◦ ( − m − ,
0) = X [ v ] (( − m − , − E + H ⋄ [ σ − m − ] = M m +1 . Bydefinition of the operator D and due to (4.6) one sees that DH ◦ = cos θ m +1 cos θ m +2 · [ 0 . . . M m +1 ∗ ∗ . . . ] ⊤ . Therefore, we havecos θ m +1 cos θ m +2 · M m +1 M m = det P m +1 ( D even D ∗ even ) / P m +1 det P m ( D even D ∗ even ) / P m . The formula M m = [ Q mk =1 cos θ k ] − det P m J / P m in (1.5) easily follows by induc-tion and since M = 1 (note that the computations given above does not requireany modification in the case m = 0 when dealing with the magnetization in evencolumns). To prove that M m also equals to | det P m U even P m | , note that( D even D ∗ even ) / = D ∗ even U ∗ even and P m D ∗ even = P m D ∗ even P m , which impliesdet P m ( D even D ∗ even ) / P m = | det P m U ∗ even P m | · | det P m D ∗ even P m | = | det P m U even P m | · Q mk =1 cos θ k . Finally, to prove the last identity in (1.5), note thatdet P m J / P m = det[ h J / f p , f q i ] m − p,q =0 det[ h f p , f q i ] m − p,q =0 for all bases f , . . . , f m − of the m -dimensional space Ran P m . Choosing the ba-sis 1 , λ, . . . , λ m − in the spectral representation of the operator J in the space L ( ν J ( dλ )) one obtains the identitydet P m J / P m = H m [ λ / ν J ]H m [ ν J ] and, similarly, det P m JP m = H m [ λν J ]H m [ ν J ]As det P m JP m = [det P m D ∗ even P m ] = [ Q nk =1 cos θ k ] , this completes the proof. (cid:3) Boundary magnetic field and the wetting phase transition.
In thissection we assume that θ k = θ < π for all k ≥
2, i.e., that we work with a fullyhomogeneous subcritical model but we allow the first interaction constant to havea different value. This can be trivially reformulated as inducing an additional mag-netic field at the first column whose strength h = 2 J corresponds to θ via (2.2).The main result is the following theorem which translates the abstract formula (1.5)into the concrete language of Toeplitz+Hankel determinants. Let q := tan θ < , r := 1 − cos θ cos θ ∈ ( − q ; 1) ,w ( z ) := | − q z | , ξ ( z ) := ( rz − q )( q z − z − q )( q z − r ) . (4.13)Note that ξ ( z ) ξ ( z − ) = 1. IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 25
Theorem 4.4.
In the setup described above, the following formula holds: M m = (1 − r ) − / det (cid:2) α k − n − β k + n + (1 − r ) / γ k + n (cid:3) m − k,n =0 , (4.14) where α s := 12 π Z π − π e − isθ w ( e iθ ) dθ , β s := 12 π Z π − π e − isθ ξ ( e iθ ) w ( e iθ ) dθ, and γ s := c · ( q /r ) s , c = ( r − q ) r − / ( r − q ) − / , if r > q and γ s := 0 otherwise.Proof. Denote a := sin θ cos θ = ( q + q − ) − . The entries of the Jacobi matrix J (see (1.4)) are given by b = (1 − r ) q − a , a = (1 − r ) / a ; b k = 1 − a , a k = a , k ≥ . Let ̺ k := (1 − rδ k, ) / , where δ k, is the Kronecker delta. The continuous spectrumof J has multiplicity 1 and equals to [1 − a , ψ k ( ζ ) := ̺ − k · [ ζ k − ξ ( ζ ) ζ − k ] , λ ( ζ ) := 1 − a · (2 + ζ + ζ − ) , ζ = e iθ , θ ∈ [0 , π ] . The coefficient ξ ( ζ ) should satisfy the condition ( b − λ ( ζ )) ψ ( ζ ) = a ψ ( ζ ) whichleads to the formula (4.13). The matrix J also has the eigenvalue λ ( ζ ) = (1 − r )( r − q ) r (1 + q ) ∈ (0 , − a ) if ζ := q /r < ξ ( ζ ) = 0. Note that ̺ k ̺ n π Z π ψ n ( e − iθ ) ψ k ( e iθ ) dθ = 12 πi I | ζ | =1 [ ζ k − n − ξ ( ζ − ) ζ k + n ] dζζ = (1 − rδ k + n, ) · δ k,n − c ζ k + n − , where c = 0 if r ≤ q and c := res z = ζ ξ ( z − ) = q (1 − r )( r − q ) r ( r − q ) if r > q . Thus, the spectral decomposition of the basis vector e n = ( δ k,n ) k ≥ reads as δ k,n = 12 π Z π ψ n ( e − iθ ) ψ k ( e iθ ) dθ + ̺ − n c ζ n − · ψ k ( ζ ) . Since λ ( e iθ ) = (1 + q ) − ( w ( e it )) , this gives the identity ̺ k ̺ n h e k , J / e n i = ̺ k ̺ n π Z π ψ n ( e − iθ ) ψ k ( e iθ ) w ( e iθ ) dθ q + c ζ k + n − ( λ ( ζ )) / = ̺ k ̺ n · [(1 + q ) − ( α k − n − β k + n ) + c ( λ ( ζ )) / ζ k + n − ] . It remains to note that the normalizing factor [ Q nk =1 cos θ k ] − in (1.5) equals to(1 − r ) − / · (1 + q ) k and hence (note also the two factors ̺ = (1 − r ) − / in thefirst row and the first column of the matrix J / ) M m = (1 − r ) − / det (cid:2) α k − n + β k + n + (1 − r ) / c · ζ k + n − (cid:3) m − k,n =0 , where c := r (1 + q ) c ( λ ( ζ )) / q (1 − r ) / = r − q r / ( r − q ) / as claimed. (cid:3) Remark free boundary conditions ) . One can pass to the limit r → − (whichcorresponds to J → + ) in the formula (4.14) since α s = α − s = β s + O (1 − r )and α = β + O ((1 − r ) ) as r → − . (It is also not hard to adapt the proofs ofTheorems 1.1 and 4.4 for this setup.) In particular, one can easily see that E + , H ⋄ [ σ ( − , ] = (1 − q ) / , r = 1 , where the sign ‘+’ in the supersciript indicates the boundary conditions at infinityand 0 stands for the value of the magnetic field h at the vertical boundary (freeboundary conditions). Note that M does not vanish at h = 0 provided that q < Remark wetting phase transition ) . Moreover, one can analytically continue theright-hand side of (4.14) to negative values of (1 − r ) / . According to [26, 50], thiscorresponds to a wetting phase transition. Informally speaking, for small negativevalues − h of the boundary magnetic field, the interface separating ‘+’ boundaryconditions at infinity from ‘ − ’ ones on the imaginary line i R touches the boundaryinfinitely often and the ‘+’ phase dominates in the bulk of the half-plane, whilefor big negative values − h this interface ‘breaks away’ from i R and the ‘ − ’ phasedominates in the bulk. For instance, one should have E + , − h H ⋄ [ σ ( − , ] = −| − r | / ( α − β ) + γ = 2 γ − E + ,h H ⋄ [ σ ( − , ]provided that h is small enough. Due to Theorem 4.4, the mismatch 2 γ disap-pears (which means that the boundary conditions at the vertical line dominatethose at infinity) if h ≥ h crit ( q ), where the critical value h crit ( q ) is specified by thecondition r = q .We refer the interested reader to [26, 50] and [40, Chapter XIII] for a discussionof this regime of the Ising model. (Note that the interpretation of the physicsbehind this effect given in the book [40] differs from the later work [26, 50].) Inparticular, [40, Fig. 13.7] suggests thatlim m →∞ E + , − h H ⋄ [ σ ( − m − , ] = (1 − q ) / for all h < h crit ( q )while, for all m ∈ N , E + , − h H ⋄ [ σ ( − m − , ] = − E + ,h H ⋄ [ σ ( − m − , ] if h ≥ h crit ( q )since γ s = 0 in the latter case. In particular, the sign of the bulk magnetizationshould flip when the negative boundary magnetic field attains the value − h crit ( q ).It would be interesting to derive this fact as well as to understand the profile of thefunction M m ( h ) in detail using Toeplitz+Hankel determinants (4.14).5. Geometric interpretation: isoradial graphs and s-embeddings
Regular homogeneous grids and isoradial graphs.
In this section webriefly discuss the geometric interpretation of the parametersexp[ − βJ h ] = x h = tan θ h , exp[ − βJ v ] = x v = tan θ v (5.1)of the homogeneous Ising model on the square grid by putting it into a more generalcontext of Z-invariant
Ising models on isoradial graphs. We refer the reader inter-ested in historical remarks on Z-invariance to the classical paper [9] due to Baxterand Enting, a standard source for the detailed treatment is [8, Sections 6 and 7].We also refer the interested reader to the paper [4] and references therein, where
IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 27 the Z-invariance was first (to the best of our knowledge) discussed in a geometriccontext, as well as to the more recent paper [41] due to Mercat. The latter paperpopularized statistical mechanics models on rhombic lattices G • ∪ G ◦ in the prob-abilistic community; the name isoradial graphs for the corresponding embeddingsof the graph G • itself was coined by Kenyon in [32] shortly afterwards. Below weadopt the notation from the recent paper [11] on this subject due to Boutillier,de Tili`ere, and Raschel and refer the interested reader to that paper for more ref-erences. The key idea of this geometric interpretation is that the combinatorialstar-triangle transforms of the Ising model (which are known as the Yang–Baxterequation in the transfer matrices context) become local rearrangements of G • ∪ G ◦ ,e.g. see [11, Fig. 5].In the notation of [11], one searches for a re-parametrization x v = x ( θ | k ) := cn( Kπ θ | k )1 + sn( Kπ θ | k ) , x h = x ( π − θ | k ) , (5.2)where cn and sn are the Jacobi elliptic functions, θ ∈ (0 , π ), k ∈ ( −∞ , K = K ( k ) is the complete elliptic integral of the first kind, see [11, Section 2.2.2].Once such a parametrization is found, it becomes useful to replace the square gridby a rectangular one, with horizontal mesh steps 2 cos θ and vertical steps 2 sin θ ,as the Ising model under consideration fits the framework of [11], with θ and π − θ being the half-angles of the rhombic lattice; note that in [11] the Ising spins areassigned to vertices of an isoradial graph while in our paper they live on faces .It is easy to see that the equations (5.1), (5.2) can be written astan θ h = sc( Kπ θ | k ) , tan θ v = sc( K − Kπ θ | k ) . In particular, the parametrization (5.2) is always possible andtan θ h tan θ v = (1 − k ) / . Furthermore, the criticality condition θ h + θ v = π is equivalent to k = 0, and M ( θ h , θ v ) = (1 − (tan θ h tan θ v ) ) / = k / if k ∈ [0 , , (5.3)the classical result of Baxter (see [8, Eq. (7.10.50)]). Moreover, the Z-invarianceallows one to treat the homogeneous Ising model on the triangular/honeycomblattices on the same foot with the model on the square grid, see [9, Fig. 2]: one has M tri ( θ tri ) = M hex ( θ hex ) = k / if x tri = x ( π | k ) , x hex = x ( π | k ) , k ≥ , where we assume that the Ising model is considered on faces of the grid and usethe same parametrization (2.2) of interaction constants as usual in our paper.The importance of the particular way to draw the lattice becomes fully trans-parent at criticality, when θ = θ h = π − θ v . (Due to Z-invariance, this conditionreads as θ tri = π or θ hex = π for the homogeneous model on faces of the triangularor honeycomb lattices.) Indeed, under the isoradial embedding, the multiplicativefactor in the asymptotics D m ∼ C σ · (2 m cos θ ) − / as m → ∞ provided by Theorem 3.9 has a clear interpretation: 2 m cos θ is nothing but the geometric distance between the two spins (located at m lattice steps from eachother) under consideration. Remark . Baxter’s formula (5.3) suggests that the spontaneous magnetizationunder criticality equals to k for the whole family of Ising models considered in [11]and not only on regular grids. Moreover, in the critical case k = 0 the asymp-totics E [ σ u σ w ] ∼ C σ · | u − w | − / as | u − w | → ∞ holds on all isoradial graphs,with the universal multiplicative constant C σ ; see [17] for further details.5.2. S-embeddings of the layered zig-zag half-plane in the periodic case.
We now move on from classical rhombic lattices to more general and flexible setupof s-embeddings suggested in [14] (see also [33, Section 7] and [13] for more de-tails) as a tool to study critical Ising models on planar graphs. We start withdiscussing a geometric intuition behind the layered setup with periodic interactionconstants θ k = θ k +2 n and conclude by formulating questions on the asymptoticbehavior of the truncated determinants (1.5) as m → ∞ in this setup.The next lemma is a simple corollary of a general result given in [19] on thecriticality condition for the Ising model on a bi-periodic planar graph. Lemma 5.2.
Let θ k = θ k +2 n for all k ≥ and some n ≥ . The layered Isingmodel in the zig-zag (half-)plane with the interaction constants x k = tan θ k betweenthe ( k − -th and k -th columns is critical (see [19] for a precise definition) if andonly if the following condition holds: Q nk =1 tan θ k = 1 . (5.4) Proof.
According to [19, Theorem 1.1], the criticality condition reads as P P ∈E ( G ) x ( P ) = P P ∈E ( G ) x ( P ) , where G denotes the fundamental domain of the grid drawn on the torus , E ( G ) isthe set of even subgraphs of G having the homology type (0 ,
0) modulo 2, and E ( G )is the set of all other even subgraphs of G (i.e., those having the types (0 , , ,
1) modulo 2). In our setup, the fundamental domain consists of 2 n verticesand one easily sees that each even subgraph P of G either contains 0 or 2 edgeslinking the k -th and the ( k + 1)-th vertices, for all k = 1 , . . . , n , or contains exactlyone of the two edges between these vertices, for all k = 1 , . . . , n . Therefore, P P ∈E ( G ) x ( P ) − P P ∈E ( G ) x ( P ) = Q nk =1 (1 − x k ) − Q nk =1 (2 x k ) . Since tan θ k = 2 x k / (1 − x k ), the claim easily follows. (cid:3) Recall that the same condition (5.4) describes the fact that the spectrum of thenon-negative Jacobi matrix J begins at 0. In this case, it is easy to see that theunique (up to a multiplicative constant) periodic solution to the equation Jψ ◦ = 0(in other words, a generalized eigenfunction corresponding to λ = 0) is given by ψ ◦ k = (sin θ k − ) − · Q k − p =1 cot θ p , k ≥ . (5.5)Our next goal is to construct an s-embedding of the bi-periodic planar Isingmodel under consideration as explained in [14, Section 6.4] (see also [33, Lemma 11]and [13] for more details). For k ∈ N and s ∈ Z , let S (( − k − , s )) = ( − t • k , s ) if k + s Z , S (( − k − , s )) = ( − t ◦ k , s ) if k + s ∈ Z , where t ◦ < t • < t ◦ < t • < . . . and t • < t ◦ < t • < t ◦ < . . . , see Fig. 6. Since thequadrilaterals with vertices ( − t • k , s ), ( − t ◦ k , s +1), ( − t • k +1 , s +1), ( − t ◦ k +1 , s ) should be IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 29 t- t- t- t- t- t- t- t- t- t- t- t- B S Φ Φ Φ Φ Φ Figure 6.
S-embedding of a periodic critical layered Ising model,see [14, Section 5]. The slopes φ k are uniquely determined by therecurrence (5.6) and the condition (5.7) coming from the requiredperiodicity of the function L S in the horizontal direction.tangential, we have t • k +1 − t ◦ k = [tan φ k +1 + tan φ k ] ,t ◦ k +1 − t • k = [cot φ k +1 + cot φ k ] , where φ k := arccot( t ◦ k − t • k ) ∈ (0 , π ) . Moreover, the formula [14, Eq. (6.3)] for the value of the Ising interaction parametergives the recurrence relationtan φ k +1 = tan θ k +1 · tan φ k , k ∈ N . (5.6)Finally, the condition that the function L S is periodic (in the horizontal direction)reads as P nk =1 tan φ k = P nk =1 cot φ k . (5.7)It is easy to see that (5.6) and (5.7) define the angles φ k uniquely and that the width of the horizontal period B S := t • k +2 n − t • k = t ◦ k +2 n − t ◦ k , k ∈ N , of thus constructed s-embedding S of the zig-zag half-plane H ⋄ equals to B S = (cid:2) P nk =1 tan φ k + P nk =1 cot φ k (cid:3) = (cid:2) P nk =1 tan φ k · P nk =1 cot φ k (cid:3) / = (cid:2) P nk =1 Q kp =1 tan θ p · P nk =1 Q kp =1 cot θ p (cid:3) / . A straightforward computation based upon (5.5) shows that this expression coin-cides with the formula (1.6) for the coefficient C J in the asymptotics of the inte-grated density of states of the matrix J at 0. More precisely, one has P nk =1 ( ψ ◦ k ) = P nk =1 (cid:2) sin − θ k − Q k − p =1 cot θ p (cid:3) = P nk =1 Q k − p =1 cot θ p , P nk =1 ( a k ψ ◦ k ψ ◦ k +1 ) − = P nk =1 (cid:2) cos − θ k Q k − p =1 tan θ p (cid:3) = P nk =1 Q kp =1 tan θ p , and therefore n − B S = (cid:2) n − P nk =1 ( ψ ◦ k ) · P nk =1 ( a k ψ ◦ k ψ ◦ k +1 ) − (cid:3) / = C J . (5.8)We conclude this section by coming back to the discussion of the link betweenthe spectral properties of the matrix J and the asymptotic behavior of the magne-tization M m as m → ∞ . Contrary to the classical isoradial setup, in the periodiclayered case we do not expect a regular behavior M m ∼ const · m − / uniformlyover all m . Instead, one should expect an oscillating prefactor A p depending onthe ‘type’ of the column under consideration: M nm + q ∼ A p · / C σ ( B S m ) − / for 1 ≤ q ≤ n and m → ∞ , where the main factor 2 / C σ ( B S m ) − / is universal and accounts the geometry ofthe s-embedding, cf. (5.8) and the asymptotics (A.9) in the homogeneous case. Notethat such oscillating behavior of (1.5) is fully consistent with the fact that supp ν J has n bands in the periodic setup instead of a single segment in the homogeneouscase. From our perspective, it would be interesting • to justify the oscillatory behavior described above and, especially, to findspectral and geometric interpretations of the coefficients A q ; • to find a natural definition of the average magnetization over the period M m = M m ( M nm +1 , . . . , M n ( m +1) ) such that M m ∼ / C σ ( B S m ) − / as m → ∞ (in other words, to find a natural average that makes 1 out of A , . . . , A n ).A. Appendix. Critical Ising model θ h = θ v = π : diagonal correlationsand the half-plane magnetization via Legendre polynomials In this appendix we work with the fully homogeneous critical (i.e., θ h = θ v = π )Ising model on the π - rotated square grid of mesh size √
2. (Note that this setupis actually more similar to Section 4 rather than to Section 3.) We begin witha discussion of the famous result of Wu (see Theorem A.3 below) that providesan explicit expression of the diagonal spin-spin correlations in terms of factorials.Using the same approach as in the core part of our paper, we give a short proof ofthis theorem by reducing the computation to the norms of the classical
Legendrepolynomials . (As communicated to the authors by J.H.H. Perk, a similar linkwith Legendre functions and Wronskian identities was the starting point of theirjoint with H. Au-Yang treatment [49] of the two-point correlations at criticality viaquadratic identities from [47]; see also Remark 3.10.) After this, we move to themagnetization M m in the (2 m )-th column of the zig-zag half-plane H ⋄ and notethat it admits a similar explicit representation via factorials (see Theorem A.4) dueto a simple Schwarz reflection argument, an identity which appears to be new.
IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 31
Remark
A.1 . The interested reader is also referred to [12, Section 3] where thenon-critical case θ = θ h = θ v < π is handled in the same way, via the OPUCpolynomials corresponding to the weight w q ( t ) = | − q e it | with q := tan θ <
1. Itwould be interesting to understand the precise link between asymptotics of theseorthogonal polynomials obtained by Basor, Chen and Haq in [5] and asymptoticsof the diagonal Ising correlations obtained by Perk and Au-Yang in [48].Let n ∈ N and assume that the π -rotated square grid is shifted so that itsvertices (resp., centers of faces) form the lattice ( − n − + k, s ) (resp., ( n + + k, s ))with k, s ∈ Z and k + s ∈ Z . Let D n := E [ σ ( − n + , σ ( n + , ]be the (infinite-volume limit of the) diagonal spin-spin correlation at distance of n diagonal steps. Denote v := ( − n − , u := ( n + ,
0) and let V ( k, s ) := X [ v , u ] (( k, s )) , k, s ∈ Z , k + s + n ∈ Z , recall that V is a spinor on the double covers branching over v and u . It followsfrom Proposition 2.2 (or, equivalently, Proposition 2.4) that V satisfies the standarddiscrete harmonicity condition [∆ V ]( k, s ) = 0 for all k, s except at the points ( ± n, V ]( k, s ) := − V ( k, s )+ [ V ( k − , s −
1) + V ( k +1 , s −
1) + V ( k − , s +1) + V ( k +1 , s +1)] . It directly follows from the definition of the observable X [ v , u ] and the self-dualityof the critical model that V ( − n,
0) = V ( n,
0) = D n . (A.1)Moreover, a straightforward computation similar to the proof of Proposition 2.2implies that [∆ V ]( ± n,
0) = − D n +1 if n ≥ , [∆ V ](0 ,
0) = − D if n = 0 . (A.2)Applying the optional stopping theorem as in the proof of Lemma 3.1, it is easy tosee that the uniformly bounded discrete harmonic spinor V is uniquely defined by itsvalues (A.1) near the branchings. Following exactly the same route as in Section 3.2we now construct V explicitly; a similar idea was used in [27, Appendix A] toconstruct the harmonic measure of the tip in the slit plane, which can be viewedas an analogue of the function V ( k − n, s ) for n = ∞ . Lemma A.2.
Let P n ( x ) := (2 n n !) − ddx [( x − n ] be the n -th Legendre polynomial.Then, for all k ∈ Z and s ∈ N such that n + k + s ∈ Z , one has V ( k, ± s ) = C n π Z π − π e − ikt ( y ( t )) s P n (cos t ) dt, (A.3) where y ( t ) = (1 − | sin t | ) / cos t and C n is chosen so that V ( ± n,
0) = D n .Proof. It is easy to see that • the values V ( k, s ) defined by (A.3) are uniformly bounded since | y ( t ) | ≤ • [∆ V ]( k, s ) = 0 if s = 0 since y ( t ) = cos t · (1 + ( y ( t )) ); • V ( k,
0) = 0 if | k | > n , thus one can view (A.3) as a function (spinor)defined on the double cover branching over v and u and vanishing overthe real line outside the segment [ v , u ], this spinor satisfies the discreteharmonicity property at ( k,
0) with | k | > n due to symmetry reasons.Moreover, the orthogonality in L ([ − , P n ( x ) to all monomials 1 , x, . . . , x n − gives − [∆ V ]( k,
0) = V ( k, − [ V ( k − ,
1) + V ( k +1 , C n π Z π − π e − ikt (1 − y ( t ) cos t ) P n (cos t ) dt = C n π Z π − π cos( kt ) | sin t | P n (cos t ) dt = C n π Z − T | k | ( x ) P n ( x ) dx = 0for all | k | < n , where T k ( x ) := cos( k arccos x ) are the Chebyshev polynomials.Therefore, the Kadanoff–Ceva fermion X [ v , u ] (( k, s )) must coincide with the right-hand side of (A.3) up to a multiplicative constant. (cid:3) The following theorem can be obtained as a simple corollary of Lemma A.2.
Theorem A.3 ( Wu).
The following explicit formula is fulfilled: D n = (cid:18) π (cid:19) n · n − Y k =1 (cid:18) − k (cid:19) k − n , n ≥ . (A.4) Proof.
Denote by p n := (2 n n !) − · (2 n )! /n ! the leading coefficient of the Legendrepolynomial P n and let t n := 2 n − , n ≥ T n , note that the value t = 1 does not match the general case. It followsfrom (A.3) that D n = C n · − n p n . On the other hand, − [∆ V ]( ± n,
0) = C n π Z − T n ( x ) P n ( x ) dx = C n t n πp n · k P n k L ([ − , = 2 C n t n π (2 n +1) p n . Due to (A.2), we conclude that for all n ≥ D n +1 D n = 2 n +1 C n π (2 n +1) p n = 2 n +1 π (2 n +1) p n = 2 π · ((2 n )!!) (2 n − n +1)!! . This easily gives (A.4) by induction. (cid:3)
We now move on to an explicit expression for the magnetization in the (2 m )-thcolumn of the zig-zag half-plane H ⋄ with ‘+’ boundary conditions: M m := E + H ⋄ [ σ ( − m − , ] . Theorem A.4.
The following identities are fulfilled for all m ∈ N : M m +1 M m = D m +2 D m +1 , M m = (cid:18) π (cid:19) m · m − Y k =1 (cid:18) − k (cid:19) ⌊ k ⌋− m . (A.5) Proof.
Similarly to Section 4.1, let v = ( − m − ,
0) and H ( − k, s ) := X [ v ] (( − k, s )) , k ∈ N , s ∈ Z , k + s Z IG-ZAG LAYERED ISING MODEL AND ORTHOGONAL POLYNOMIALS 33 be the half-plane fermionic observable. This is a bounded discrete harmonic (exceptat ( − m − , H ⋄ branching over v which satisfythe boundary conditions H (0 , s ) = 2 − / · [ H ( − , s −
1) + H ( − , s +1)] , s Z , (A.6)on the imaginary line (see (4.3) and (4.4)). Denote V ( ± k, s ) := CH ( k, s ) if k ∈ N ,V (0 , s ) := 2 − / · CH (0 , s ) if k = 0 , s ∈ Z , k + s Z , (A.7)where C := D m +1 /M m ; up to a change of the multiplicative normalization, thisis nothing but the extension of H from the left half-plane to the full plane via thediscrete Schwartz reflection. By construction, V is a spinor on the double coverof the full-plane branching over v and u := (2 m + ,
0) which is discrete harmoniceverywhere (including points on the imaginary line) except at points ( ± (2 m +1) , V ( ± (2 m + 1) ,
0) = D m +1 . Therefore, it co-incides with the full-plane observable X [ v , u ] (( k, s )) discussed above. In particular,(A.7) implies the identity D m +2 = − [∆ V ]( − m − ,
0) = − C · [∆ H ]( − m − ,
0) = C · M m +1 which is equivalent to the first identity in (A.5). The explicit formula for M m easilyfollows from the explicit formula (A.4) by induction. (cid:3) Remark
A.5 . Similarly, let M m − denote the magnetization in the (2 m − M m + / M m − = D m +1 / D m , m ∈ N , where we formally set M − := √
2, this convention is the result of the additionalfactor relating the values of the half-plane and the full-plane observables on theimaginary line via (A.7). By induction, one easily gets the identity M m + M m = √ · D m +1 , m ∈ N , (A.8)and an explicit formula for M m + , which is similar to (A.5). Finally, a straightfor-ward analysis gives the asymptotics D n ∼ C σ · (2 n ) − / , M m ∼ / C σ · (2 m ) − / , n, m → ∞ , (A.9)where C σ = 2 e ζ ′ ( − is the same universal constant as in Theorem 3.9. Notethat we prefer to encapsulate the factors 2 n and 2 m (rather than simply n and m ),respectively, as they are equal to the geometric distance between the two spinsunder consideration and the distance from the spin σ ( − m − , to the boundary ofthe half-plane H ⋄ , respectively. References [1] Helen Au-Yang. Criticality in alternating layered Ising models. II. Exact scaling theory.
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