NUPHB114682114682S0550-3213(19)30168-310.1016/j.nuclphysb.2019.114682The AuthorQuantum Field Theory and Statistical SystemsPermutation invariant Gaussian matrix modelsSanjayeRamgoolamab⁎s.ramgoolam@qmul.ac.ukaCentre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UKCentre for Research in String TheorySchool of Physics and AstronomyQueen Mary University of LondonMile End RoadLondonE1 4NSUKCentre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UKbNational Institute for Theoretical Physics, School of Physics and Centre for Theoretical Physics, University of the Witwatersrand, Wits, 2050, South AfricaNational Institute for Theoretical PhysicsSchool of PhysicsCentre for Theoretical PhysicsUniversity of the WitwatersrandWits2050South AfricaNational Institute for Theoretical Physics, School of Physics and Centre for Theoretical Physics, University of the Witwatersrand, Wits, 2050, South Africa⁎Correspondence to: Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK.Centre for Research in String TheorySchool of Physics and AstronomyQueen Mary University of LondonMile End RoadLondonE1 4NSUKEditor: Hubert SaleurAbstractPermutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group SD where D is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.1IntroductionIn the context of distributional semantics [1,2], the meaning of words is represented by vectors which are constructed from the co-occurrences of a word of interest with a set of context words. In tensorial compositional distributional semantics [3–7], different types of words, depending on their grammatical role, are associated with vectors, matrices or higher rank tensors. In [8,9] we initiated a study of the statistics of these tensors in the framework of matrix/tensor models. We focused on matrices associated with adjectives or verbs, constructed by a linear regression method, from the vectors for nouns and for adjective-noun composites or verb-noun composites.We developed a 5-parameter Gaussian model,(1.1)Z(Λ,a,b,J0,JS)=∫dMe−Λ2∑i=1DMii2−14(a+b)∑i<j(Mij2+Mji2)e−12(a−b)∑i<jMijMji+J0∑iMii+JS∑i<j(Mij+Mji). The parameters JS,J0,a,b,Λ are coefficients of five linearly independent linear and quadratic functions of the D2 random matrix variables Mi,j which are permutation invariant, i.e. obey the equation(1.2)f(Mi,j)=f(Mσ(i),σ(j)) for σ∈SD, the symmetric group of all permutations of D distinct objects. This SD invariance implements the notion that the meaning represented by the word-matrices is independent of the ordering of the D context words. General observables of the model are polynomials f(M) obeying the condition (1.2). At quadratic order there are 11 linearly independent polynomials, which are listed in Appendix B of [8]. A three dimensional subspace of quadratic invariants was used in the model above. The most general Gaussian matrix model compatible with SD symmetry considers all the eleven parameters and allows coefficients for each of them. What makes the 5-parameter model relatively easy to handle is that the diagonal variables Mii are each decoupled from each other and from the off-diagonal elements, and there are D(D−1)/2 pairs of off-diagonal elements. For each i<j, Mij and Mji mix with each other so the solution of the model requires an inversion of a 2×2 matrix.Expectation values of f(M) are computed as(1.3)〈f(M)〉≡1Z∫dMf(M)EXP where EXP is the product of exponentials in (1.1).Representation theory of SD offers the techniques to solve the general permutation invariant Gaussian model. The D2 matrix elements Mij transform as the tensor product VD⊗VD of two copies of the natural representation VD. We first decompose VD⊗VD into irreducible representations of the diagonal SD.(1.4)VD⊗VD=2V0⊕3VH⊕V2⊕V3 The trivial (one-dimensional) representation V0 occurs with multiplicity 2. The (D−1)-dimensional irreducible representation (irrep) VH occurs with multiplicity 3. V2 is an irrep of dimension (D−1)(D−3)2 which occurs with multiplicity 1. Likewise, V3 of dimension (D−1)(D−2)2 occurs with multiplicity 1. As a result of these multiplicities, the 11 parameters can be decomposed as(1.5)11=1+1+3+6 3 is the size of a symmetric 2×2 matrix. 6 is the size of a symmetric 3×3 matrix. More precisely the parameters form(1.6)M=R+×R+×M2+×M3+ where R+ is the set of real numbers greater or equal to zero, Mr+ is the space of positive semi-definite matrices of size r. Calculating the correlators of this Gaussian model amounts to inverting a symmetric 2×2 matrix, inverting a symmetric 3×3 matrix, and applying Wick contraction rules, as in quantum field theory, for calculating correlators. There is a graph basis for permutation invariant functions of M. This is explained in Appendix B of [8] which gives examples of graph basis invariants and representation theoretic counting formulae which make contact with the sequence A052171 - directed multi-graphs with loops on any number of nodes - of the Online Encyclopaedia of Integer Sequences (OEIS) [10].In this paper we show how all the linear and quadratic moments of the graph-basis invariants are expressed in terms of the representation theoretic parameters of (1.6). We also show how some cubic and quartic graph basis invariants are expressed in terms of these parameters. These results are analytic expressions valid for all D.The paper is organised as follows. Section 2 introduces the relevant facts from the representation theory of SD we need in a fairly self-contained way, which can be read with little prior familiarity of rep theory, but only knowledge of linear algebra. This is used to define the 13-parameter family of Gaussian models (equations (2.71), (2.72), (2.73)). Section 3 calculates the expectation values of linear and quadratic graph-basis invariants in the Gaussian model. Sections 4 and 5 describe calculations of expectation values of a selection of cubic and quartic graph-basis invariants in the model.2General permutation invariant Gaussian Matrix modelsWe solved a permutation invariant Gaussian Matrix model with 2 linear and 3 quadratic parameters [8], obtaining analytic expressions for low order moments of permutation invariant polynomial functions of a matrix variable as a function of the 5 parameters (section 6 of [8]). The linear parameters are coefficients of linear permutation invariant functions of M and the quadratic parameters (denoted Λ,a,b) are coefficients of quadratic functions. We explained the existence of a 2+11 parameter family of models, based on the fact that there are 11 linearly independent quadratic permutation invariant functions of a matrix. The general 2+11-parameter family of models can be solved with the help of techniques from the representation theory of SD.We give a brief informal description of the key concepts we will use here. Further information can be found in [11–14], and we will give more precise references below. A representation of a finite group G is a pair (V,DV) consisting of a vector space V and a homomorphism DV from G to the space of invertible linear operators acting on V. Physicists often speak of a representation V of G, when the accompanying homomorphism is left implicit. The homomorphism associates to each g∈G a linear operator DV(g). Distinguished among the representations of G are the irreducible representations (irreps). It is known that any representation of G is isomorphic to a direct sum of irreducible representations. For further explanations of these statements see Lecture 1 of [12]. When a representation V is a direct sum of V1,V2,⋯,Vk, we express this as(2.1)V=V1⊕V2⊕⋯⊕Vk. This implies that the linear operators DV(g) corresponding to group elements g∈G can, after an appropriate choice of basis in V, be put in a block diagonal form where the blocks are DV1(g),DV2(g),⋯,DVk(g). The problem of finding this change of basis is called “reducing the representation V into a direct sum of irreducibles”.Given two representations (V1,DV1) and (V2,DV2) of G, the tensor product space V1⊗V2 is a representation of the product group G×G, which consists of pairs (g1,g2) with g1,g2∈G. The product group G×G has a subgroup of pairs (g,g) which is called the diagonal subgroup of G, denoted Diag(G). The tensor product space V1⊗V2 is also a representation of this diagonal subgroup (see for example Chapter 1 of [14]). The linear transformation which reduces V1⊗V2 into a direct sum of irreducibles of Diag(G) is called the Clebsch-Gordan decomposition. The matrix elements of the transformation are called Clebsch-Gordan coefficients. More details on these can be found in Chapter 5 of [11]. These can be used to construct projection operators for the subspaces of the tensor product space corresponding to particular irreducible representations.In section 2.1, we introduce the natural representation VD of SD. We note that the space of linear combinations of the matrix variables Mij is isomorphic as a vector space to VD⊗VD. We recall the known fact that VD is isomorphic to a direct sum of two irreducible representationsVD=V0⊕VH and give the explicit change of basis which demonstrates this isomorphism. The tensor product is thus isomorphic to a direct sum(2.2)VD⊗VD=(V0⊗V0)⊕(V0⊗VH)⊕(VH⊗V0)⊕(VH⊗VH) This leads to the definition (Equation (2.23)) of SD×SD covariant variables S00,Sa0H,SaH,0,SabHH, which correspond to the four terms in the expansion (2.2).In section 2.2, we describe the space of linear combinations of Mij as a representation of Diag(SD):(2.3) Span {Mij:1≤i,j≤D}=V0⊕V0⊕VH⊕VH⊕VH⊕V2⊕V3=⨁α=12V0(α)⨁α=13VH(α)⊕V2⊕V3 The irreps V2,V3 have dimensions (D)(D−3)/2 and (D−1)(D−2)/2. The multiplicity index α keeps track of the fact the same irrep appears multiple times in the decomposition into irreducibles of Span (Mij). The isomorphism of representations of SD above implies the identity relating the dimensions(2.4)D2=2+3(D−1)+D(D−3)2+(D−1)(D−2)2 This decomposition leads to the definition, in equations (2.51), (2.52), (2.53), of variables SVi;α transforming according to the decomposition (2.3).The next key observation is to think about the vector space of quadratic polynomials in indeterminates {x1,x2,⋯,xN} in a way which is amenable to the methods of representation theory. Consider a vector space VN spanned by x1,x2,⋯,xN. The quadratic polynomials are spanned by the set of monomials xixj which contains N(N+1)/2 elements. The vector space can be identified with the subspace of the tensor product VN⊗VN which is invariant under the exchange of the two factors using the map(2.5)xixj→(xi⊗xj+xj⊗xi). This subspace of VN⊗VN is denoted by Sym2(VN). In section 2.2 we apply this observation to the space of quadratic polynomials in the matrix variables Mij. They form a vector space which is isomorphic to Sym2(VD⊗VD).Using the decomposition (2.3), we are able to find the SD invariants by using a general theorem about invariants in tensor products of irreducible representations. For two irreps VR,VS, the tensor product VR⊗VS contains the trivial representation of the diagonal SD only if R=S, i.e. VR is isomorphic to VS, and further it is also known that this invariant appears in the symmetric subspace Sym2(VR)⊂(VR⊗VR). For further information on this useful fact, the reader is referred to Chapter 5 of [11].This culminates in section 2.3 in an elegant representation theoretic description of the quadratic invariants in the matrix variables, using the linear combinations SVi;α. With this description in hand, we introduce a set of representation theoretic parameters for the 13-parameter Gaussian matrix models, see equations (2.71) and (2.72). In terms of these parameters, the linear and quadratic expectation values of SVi;α are simple (see equations (2.75), (2.76), (2.77)). The computation of the correlators of low order polynomial invariant functions of the matrices then follows using Wick's theorem from quantum field theory (see for example Appendix A of [25]).2.1Matrix variables Mij and the natural representation of SD×SDThe matrix elements Mij, where i,j run over {1,2,⋯,D} span a vector space of dimension D2. It is isomorphic to the tensor product VD⊗VD, where VD is a D-dimensional space. Consider VD as a span of D basis vectors {e1,e2,⋯,eD}. This vector space VD is a representation of SD. For every permutation σ∈SD, there is a linear operator ρVD(σ) defined by(2.6)ρVD(σ)ei=eσ−1(i) on the basis vectors and extended by linearity. With this definition, ρVD is a homomorphism from SD to linear operators acting on VD(2.7)ρVD(σ1)ρVD(σ2)=ρVD(σ1σ2). We introduce an inner product (.,.) where the ei form an orthonormal basis(2.8)(ei,ej)=δij. We can form the following linear combinations(2.9)E0=1D(e1+e2+⋯+eD)E1=12(e1−e2)E2=16(e1+e2−2e3)⋮Ea=1a(a+1)(e1+e2+⋯+ea−aea+1)⋮ED−1=1D(D+1)(e1+e2+⋯+eD−1−(D−1)eD). E0 is invariant under the action of SD(2.10)ρVD(σ)E0=E0 since, for any σ, we have(2.11)eσ−1(1)+eσ−1(2)+⋯+eσ−1(D)=e1+e2+⋯+eD. Thus the one-dimensional vector space spanned by E0 is an SD invariant vector subspace of VD. We can call this vector space V0. The vector space spanned by Ea, where 1≤a≤(D−1), which we call VH, is also an SD-invariant subspace(2.12)ρVD(σ)Ea∈VH. We have a matrix DH(σ) with matrix elements DabH(σ) such that(2.13)ρVD(σ)Ea=∑b=1D−1DbaH(σ)Eb. These matrices are obtained by using the action on the ei and the change of basis coefficients. The vectors EA for 0≤A≤D−1 are orthonormal under the inner product (2.8)(2.14)(EA,EB)=δA,B. All the above facts are summarised by saying that the natural representation VD of SD decomposes as an orthogonal direct sum of irreducible representations of SD as(2.15)VD=V0⊕VH. By reading off the coefficients in the expansion of the E0,Ea in VH, we can define the coefficients(2.16)C0,i=(E0,ei)Ca,i=(Ea,ei) using the inner product (2.8). They are(2.17)C0,i=1DCa,i=Na(−aδi,a+1+∑j=1aδji)Na=1a(a+1). The orthonormality means that(2.18)∑i=1DC0,iC0,i=1∑i=1DCa,iCb,i=δa,b∑i=1DC0,iCa,i=0. The last equation implies that(2.19)∑i=1DCa,i=0. From(2.20)∑A=0D−1CA,iCA,j=C0,iC0,i+∑a=1D−1Ca,iCa,j=δi,j we deduce(2.21)∑a=1D−1Ca,iCa,j=(δij−1D)≡F(i,j). As we will see, this function F(i,j) will play an important role in calculations of correlators in the Gaussian model. It is the projector in VD for the subspace VH, obeying(2.22)∑j=1DF(i,j)F(j,k)=F(i,k)∑i=1DF(i,i)=(D−1).Now we will use these coefficients CA,i to build linear combinations of the matrix elements Mi,j which have well-defined transformation properties under SD×SD. Define(2.23)S00=∑i,j=1DC0,iC0,jMij=1D∑i,j=1DMijSa0H=∑i,j=1DC0,iCa,jMij=1D∑i,j=1DCa,jMijSaH0=∑i,j=1DCa,iC0,jMij=1D∑i,j=1DCa,iMijSabHH=∑i,j=1DCa,iCb,jMi,j. The a,b indices range over 1⋯(D−1). These variables are irreducible under SD×SD, transforming as V0⊗V0,V0⊗VH,VH⊗V0,VH⊗VH. Under the diagonal SD, the first three transform as V0,VH,VH while SabHH form a reducible representation.Conversely, we can write these M variables in terms of the S variables, using the orthogonality properties of the C0,i,Ca,i,(2.24)Mij=C0,iC0,jS00+∑a=1D−1C0,iCa,jSa0H+∑a=1D−1Ca,iC0,jSaH0+∑a,b=1D−1Ca,iCb,jSHHab=1DS00+1D∑a=1D−1Ca,jSa0H+1D∑a=1D−1Ca,iSaH0+∑a,b=1D−1Ca,iCb,jSabHH.The next step is to consider quadratic products of these S-variables, and identify the products which are invariant. In order to do this we need to understand the transformation properties of the above S variables in terms of the diagonal action of SD. It is easy to see that S00 is invariant. Sa0H and SaH0 both have a single a index running over {1,2,⋯,(D−1)}, and they transform in the same way as VH. The vector space spanned by SabHH form a space of dimension (D−1)2 which is(2.25)VH⊗VH. Permutations act on this as(2.26)σ(SabHH)=∑a1,b1Da1aH(σ)Db1bH(σ)Sa1,b1HH, using the matrices DH(σ) introduced in (2.13).2.2Decomposition of matrix variables as irreducible representations of Diag(SD)⊂SD×SDIn this section we will perform a further change of variables to introduce variables SVi;α which transform according to irreps of the diagonal subgroup Diag(SD)⊂SD×SD.•The representation space VH⊗VH can be decomposed into irreducible representations (irreps) of the diagonal SD action as(2.27)VH⊗VH=V0⊕VH⊕V2⊕V3. In Young diagram notation for irreps of SD, listing the row lengths of the Young diagram, we have(2.28)V0→[D]VH→[D−1,1]V2→[D−2,2]V3→[D−2,1,1].•These irreps are known to have dimensions 1,(D−1),D(D−3)2,(D−1)(D−2)2. They add up to (D−1)2 which is the dimension of VH⊗2.•The vector ∑a=1D−1Ea⊗Ea is invariant under the diagonal action of σ on VH⊗VH. Using the fact that VH is a subspace of VD described by the coefficients Ca,i defined in (2.16), the action of σ on VH is given by(2.29)DabH(σ)=(Ea,σEb)=∑i=1DCa,iCb,σ(i). These can be verified to satisfy the homomorphism property(2.30)∑b=1D−1DabH(σ)DbcH(τ)=DacH(στ). We also have DabH(σ−1)=DbaH(σ). Using these properties, we can show that ∑aEa⊗Ea is invariant under the diagonal action. The vector(2.31)1D−1∑a=1D−1Ea⊗Ea has unit norm, using the inner product on VD⊗VD obtained from (2.8), and defines a normalized vector in the V0 subspace of the direct sum decomposition of VH⊗VH given in (2.27). From this expression, we can read off the Clebsch-Gordan coefficients for the trivial representation V0 in VH⊗VH(2.32)Ca,bH,H→V0=δabD−1. Using these we define SHH→V0 as(2.33)SHH→V0=∑a,b=1D−1Ca,bH,H→V0SabHH=1D−1∑a=1D−1SaaHH•The vectors in the VH subspace on the RHS of the direct sum decomposition (2.27) are some linear combinations(2.34)∑b,c=1D−1Cb,c;aH,H→HSbcHH≡SaH,H→H. The coefficients Cb,c;aH,H→H are some representation theoretic numbers (called Clebsch-Gordan coefficients) which satisfy the orthonormality condition(2.35)∑b,c=1D−1Cb,c;aH,H→HCb,c;dH,H→H=δa,d.As shown in Appendix B, these Clebsch-Gordan coefficients are proportional to Ca,b,c≡∑iCa,iCb,iCc,i(2.36)Ca,b;cHH→H=D(D−2)Ca,b,c. It is a useful fact that the Clebsch-Gordan coefficients for VH⊗VH→VH can be usefully written in terms of the Ca,i describing VH as a subspace of the natural representation. This has recently played a role in the explicit description of a ring structure on primary fields of free scalar conformal field theory [15]. It would be interesting to explore the more general construction of explicit Clebsch-Gordan coefficients and projectors in the representation theory of SD in terms of the Ca,i.•Similarly for V2,V3 we have corresponding vectors and Clebsch-Gordan coefficients(2.37)∑b,c=1D−1Cb,c;aH,H→V2Sb,c≡SaHH→V2 where a ranges from 1 to Dim(V2)=D(D−3)2. We have the orthogonality property(2.38)∑b,c=1D−1Cb,c;a1H,H→V2Cb,c;a2H,H→V2=δa1,a2. And for V3(2.39)∑b,c=1D−1Cb,c;aH,H→V3SbcHH≡SaHH→V3∑b,c=1D−1Cb,c;a1H,H→V3Cb,c;a2H,H→V3=δa1,a2. Here the a,a1,a2 runs over 1 to (D−1)(D−2)2.•The projector for the subspace of VH⊗VH transforming as VH under the diagonal SD is(2.40)(PH,H→H)a,b;c,d=∑e=1D−1Ca,b;eH,H→HCH,H→Hc,d;e=D(D−2)∑e=1D−1Ca,b,eCc,d,e.•The projector PH,H→V0 for V0 in VH⊗VH is(2.41)(PH,H→V0)a,b;c,d=1(D−1)δa,bδc,d. V[n−2,1,1]=V3 is just the anti-symmetric of VH⊗VH. It is the orthogonal complement to VH⊕V0 inside the symmetric subspace of VH⊗VH which is invariant under the swop of the two factors (often denoted Sym2(VH))(2.42)(PH,H→V2)=(1−PH,H→H−PH,H→V0)(1+s)2 where the swop s acting on VH⊗VH has matrix elements(2.43)(s)a,b;c,d=δa,dδb,c. The quadratic invariant corresponding to V2 is(2.44)SabHH(PH,H→V2)a,b;c,dScdHH. The quadratic invariant corresponding to V3 is similar. We just have to calculate(2.45)PH,H→V3=12(1−s).•The inner product(2.46)〈Mij,Mkl〉=δikδjl is invariant under the action σ(Mi,j)=Mσ−1(i),σ−1(j).(2.47)〈σ(Mij),σ(Mkl)〉=〈Mij,Mkl〉.•The following is an important fact about invariants. Every irreducible representation of SD, let us denote it by VR has the property that(2.48)Sym2(VR) contains the trivial irrep once. This invariant is formed by taking the sum over an orthonormal basis ∑AeAV⊗eVA. The invariance is proved as follows(2.49)DV⊗V(σ)∑AeAV⊗eAV=∑ADV(σ)eAV⊗DV(σ)eVA=∑A∑B,CDBAV(σ)DCAV(σ)eBV⊗eCV=∑A,B,CDBAV(σ)DACV(σ−1)eBV⊗eCV=∑B,CδB,CeBV⊗eCV=∑AeAV⊗eAV. In the first equality we have used the definition of the diagonal action of σ on the tensor product space.•To summarize the matrix variables Mij can be linearly transformed to the following variables, organised according to representations of the diagonal SD(2.50) Trivial rep:S00,SHH→V0 Hook rep:Sa0H,SaH0,SaHH→H The rep V2:SaHH→V2 The rep V3:SaHH→V3.•For convenience, we will also use simpler names(2.51)SV0;1=S00SV0;2=SHH→V0 where we introduced labels 1,2 to distinguish two occurrences of the trivial irrep V0 in the space spanned by the Mij. The variables S0,0,SH,H→V0 were first introduced in (2.23) and (2.33) respectively. We will also use(2.52)SaH;1=Sa0,H→H≡Sa0HSaH;2=SaH,0→H≡SaH0SaH;3=SaH,H→H where we introduced labels 1,2,3 to distinguish the three occurrences of VH in the space spanned by Mij. The variables Sa0H,SaH0 were introduced earlier in (2.23). For the multiplicity-free cases, we introduce(2.53)SaV2=SaHH→V2SaV3=SaHH→V3.The Mij variables can be written as linear combinations of the S variables. Rep-basis expansion of Mij is(2.54)Mij=C0,iC0,jS00+∑a,b=1D−1Ca,iCb,jSabHH+∑a=1D−1C0,iCa,jSa0H+∑a=1D−1Ca,iC0,jSaH0=1DS00+1D∑a=1D−1Ca,jSa0H+1D∑a=1D−1Ca,iSaH0+∑a,b=1D−1Ca,iCb,jSabHH=1DS00+1D∑a=1D−1Ca,jSa0H+1D∑a=1D−1Ca,iSaH0+∑a,bCa,iCb,j∑V∈{V0,VH,V2,V3}∑c=1DimVCa,b;cHH→VScHH→V=1DS00+1D∑a=1D−1Ca,jSa0H+1D∑a=1D−1Ca,iSaH0+1D−1∑a=1D−1Ca,iCa,jSHH→V0+∑a,b=1D−1Ca,iCb,j∑c=1D−1Ca,b;cHH→HSHH→Hc+∑a,b=1D−1Ca,iCb,j∑c=1DimV2Ca,b;cHH→V2ScHH→V2+∑a,b=1D−1∑c=1DimV3Ca,iCb,jCa,b;cHH→V3ScHH→V3. In going from first to second line, we have used the fact that the transition from the natural representation to the trivial representation is given by simple constant coefficients(2.55)C0,j=1D. In the third line, we have used the Clebsch-Gordan coefficients for VH⊗VH→V, obeying the orthogonality(2.56)∑a,b=1D−1CabcH,H→VCabc′H,H→V=δcc′. For V=V0, which is one dimensional, we just have(2.57)CabHH→V0=δabD−1, in accordance with (2.31). The index c ranges over a set of orthonormal basis vectors for the irrep V, i.e. extends over a range equal to the dimension of V, denoted DimV. It is now useful to collect together the terms corresponding to each irrep V0,VH,V2,V3(2.58)Mij=(1DS00+1D−1∑a=1D−1Ca,iCa,jSHH→0)+(1D∑a=1D−1Ca,jSa0H+1D∑a=1D−1Ca,iSaH0+∑a,b,c=1D−1Ca,iCb,jCa,b;cHH→HScHH→H)+∑a,b=1D−1∑c=1DimV2Ca,iCb,jCa,b;cHH→V2ScHH→V2+∑a,b=1D−1∑c=1DimV3Ca,iCb,jCa,b;cHH→V3ScHH→V3. Using the notation of (2.51), (2.52), (2.53), we write this as(2.59)Mij=(1DSV0;1+1D−1∑a=1D−1Ca,iCa,jSV0;2)+(1D∑a=1D−1Ca,jSaH;1+1D∑a=1D−1Ca,iSaH;2+∑a,b,c=1D−1Ca,iCb,jCa,b;cHH→HScH;3)+∑a,b=1D−1∑c=1DimV2Ca,iCb,jCa,b;cHH→V2ScHH→V2+∑a,b=1D−1∑c=1DimV3Ca,iCb,jCa,b;cHH→V3ScHH→V3.•The discussion so far has included explicit bases for VH inside VD which are easy to write down. A key object in the above discussion is the projector F(i,j) defined in (2.21). For the irreps V2,V3 which appear in VH⊗VH, we will not need to write down explicit bases. Although Clebsch-Gordan coefficients for H,H→V2 and H,H→V3 appear in some of the above formulae, we will only need some of their orthogonality properties rather than their explicit forms. The projectors for V2,V3 in VD⊗VD can be written in terms of the F(i,j), and it is these projectors which play a role in the correlators we will be calculating.2.3Representation theoretic description of quadratic invariantsWith the above background of facts from representation theory at hand, we can give a useful description of quadratic invariants. Quadratic invariant functions of Mij form the invariant subspace of Sym2(VD⊗VD) since Mij transform as VD⊗VD.(2.60)(VD⊗VD)=(V0⊕VH)⊗(V0⊕VH)=(V00⊕V0,HH⊕VH,0H⊕VH,H0⊕VH,HH⊕VH,HV2⊕VH,HV3). So there are two copies of V0, namely V0,00,VH,H0. Sym2(V0,00⊕VH,H0) contains three invariants:(2.61)(S00)2=(SV0;1)2(S00SHH→0)=SV0;2SV0;1=SV0;1SV0;2(SHH→0)2=(SV0;2)2. These are all easy to write in terms of the original matrix variables, using the formulae for S-variables in terms of M given earlier. The relevant equations are (2.51), (2.52), (2.53) where S-variables for irreps of Diag(SD)⊂SD×SD along with multiplicity labels are introduced, and earlier equations (2.23), (2.31), (2.34), (2.37) which introduce S-variables labelled by SD×SD irreps. These latter are more directly related to the Matrix variables, but the former are needed to give an elegant description of the quadratic invariants.The general invariant quadratic function of the SV0;α variables is(2.62)∑α,β=12(ΛV0)αβSV0;αSV0;β. ΛV0 is a 2×2 symmetric matrix. As we will see later, in defining the Gaussian model, this matrix will be restricted to be positive semi-definite.There are three copies of VH, namely V0,HH,VH,0H,VH,HH. These lead to 6 invariants:(2.63)∑aSa0,H→HS0,H→Ha=∑aSaH;1SaH;1∑aSaH,0→HSH,0→Ha=∑aSaH;2SaH;2∑aSaH,H→HSaH,H→H=∑aSaH;3SaH;3∑aSa0,H→HSH,0→Ha=∑aSaH;1SaH;2=∑aSaH;2SaH;1∑aSa0,H→HSH,H→Ha=∑aSaH;1SaH;3=∑aSaH;3SaH;1∑aSH,0→HaSH,H→Ha=∑aSaH;2SaH;3=∑aSaH;3SaH;2. The sum over a runs over (D−1)=Dim(VH) elements of a basis for VH. Thus the general quadratic invariants arising from the H representation among the Mij are(2.64)∑α,β=13(ΛH)αβ∑aSaH;αSaH;β. We introduced parameters (ΛH)αβ forming a symmetric 3×3 matrix. When we define the general Gaussian measure, we will see that this matrix will be required to be a positive definite matrix.The quadratic invariants constructed from the V2,V3 variables are(2.65)(ΛV2)∑a=1DimV2SaV2SaV2(ΛV3)∑a=1DimV3SaV3SaV3. When we define the general Gaussian measure, we will take the parameters ΛV2,ΛV3 to obey ΛV2,ΛV3≥0.2.4Definition of the Gaussian modelsThe measure dM for integration over the matrix variables Mij is taken to be the Euclidean measure on RD2 parametrised by the D2 variables(2.66)dM≡∏idMii∏i≠jdMij. Since the variables SaV;α defined in (2.51), (2.52), (2.53) are given by an orthogonal change of basis, we can show that(2.67)dM=dSV0;1dSV0;2∏a=1DimVHdSaH;1dSaH;2dSaH;3∏a=1DimV2dSaV2∏a=1DimV3dSaV3. Indeed writing MA for the matrix variables, where A runs over the D2 pairs (i,j) and SB for SD-covariant variables, where B runs over all the factors in (2.67), we have(2.68)dM=∏AdMA=|detJ|∏BdSB with(2.69)JAB=∂MA∂SB. Now the SB variables are obtained from MA by an orthogonal basis change, and symmetric group properties also allow the matrix to be chosen to be real. This implies that the matrix is orthogonal(2.70)JJT=1. Hence detJ has magnitude 1, and we have the claimed identity (2.67).The model is defined by integration. The partition function is(2.71)Z(μ1,μ2;ΛV0,ΛH,ΛV2,ΛV3)=∫dMe−S where the action is a combination of linear and quadratic functions.(2.72)S=−∑α=12μαV0SV0;α+12∑α,β=12SV0;α(ΛV0)αβSV0;β+12∑a=1D−1∑α,β=13SaH;α(ΛH)αβSaH;β+12ΛV2∑a=1(D−1)(D−2)/2SaV2SaV2+12ΛV3∑a=1D(D−3)/2SaV3SaV3.The expectation values of permutation invariant polynomials f(M) are defined by(2.73)〈f(M)〉=1Z∫dMe−Sf(M). These expectation values can be computed using standard techniques from quantum field theory, specialised to matrix fields in zero space-time dimensions (see Appendix A for some explanations). Textbook discussions of these techniques are given, for example in [24], [25]. For linear functions, the non-vanishing expectation values are those of the invariant variables, which transform as V0 under the SD action(2.74)〈SV0;α〉=∑β(Λ−1)αβμβ. We introduce the definition(2.75)μ˜α≡∑β(Λ−1)αβμβ. We have defined variables μ˜1,μ˜2 for convenience. The variables transforming according to VH,V2,V3 have vanishing expectation values(2.76)〈SaH;α〉=0〈SaV2〉=0〈SaV3〉=0. The quadratic expectation values are(2.77)〈SaVi;αSbVj;β〉=〈SaVi;αSbVj;β〉conn+〈SaVi;α〉〈SbVj;β〉, where(2.78)〈SaVi;αSbVj;β〉conn=δ(Vi,Vj)(ΛVi−1)αβδab. The notation 〈..〉conn is explained in the Appendix A. The Vi,Vj can be V0,VH,V2,V3. The delta function means that these expectation values vanish unless the two irreps Vi,Vj are equal. While δab is the identity in the state space for each Vi. The fact that the mixing matrix in the multiplicity indices α,β is the inverse of the coupling matrix ΛV is a special (zero-dimensional) case of a standard result in quantum field theory, where the propagator is the inverse of the operator defining the quadratic terms in the action. The decoupling between different irreps follows because of the factorised form of the measure dMe−S in (2.71).The requirement of an SD invariant Gaussian measure has led us to define variables SV,α, transforming in irreducible representations of SD. The action is simple in terms of these variables. This is reflected in the fact that the above one and two-point functions are simple in terms of the parameters of the model.When ΛV2>0,ΛV3>0 and ΛH,ΛV0 are positive-definite real symmetric matrices (i.e. real symmetric matrices with positive eigenvalues), then the partition function Z is well defined as well as the numerators in the definition of 〈f(M)〉. We can relax these conditions, allowing ΛV2,ΛV3≥0 and ΛH,ΛV0 positive semi-definite, by appropriately restricting the f(M) we consider. For example, if ΛV2=0, we consider functions f(M) which do not depend on SV2, which ensures that the ratios defining 〈f(M)〉 are well-defined.Thus the complete set of constraints on the representation theoretic parameters are(2.79)Det(ΛV0)≥0Det(ΛVH)≥0ΛV2≥0ΛV3≥0. More explicitly(2.80)(ΛV0)11(ΛV0)22−((ΛV0)12)2≥0(ΛH)11((ΛH)22(ΛH)33−((ΛH)23)2)−(ΛH)12((ΛH)12(ΛH)33−(ΛH)23(ΛH)13)+(ΛH)13((ΛH)12(ΛH)23−(ΛH)13(ΛH)22)≥0ΛV2≥0ΛV3≥0.With these linear and quadratic expectation values of representation theoretic matrix variables S available, the expectation value of a general polynomial function of Mij can be expressed in terms of finite sums of products involving these linear and quadratic expectation values. This is an application of Wick's theorem in the context of QFT. We will explain this for the integrals at hand in Appendix A and describe the consequences of Wick's theorem explicitly for expectation values of functions up to quartic in the matrix variables. We will be particularly interested in the expectation values of polynomial functions of the Mij which are invariant under SD action and can be parametrised by graphs. While the mixing between the S variables in the quadratic action is simple, there are non-trivial couplings between the D2 variables Mij if we expand the action in terms of the M variables. This will lead to non-trivial expressions for the expectation values of the graph-basis polynomials.These expectation values were computed for the 5-parameter Gaussian model in [8]. They were referred to as theoretical expectation values 〈f(M)〉, which were compared with experimental expectation values 〈f(M)〉EXPT. These experimental expectation values were calculated by considering a list of words labelled by an index A ranging from 1 to N, and their corresponding matrices MA,(2.81)〈f(M)〉EXPT=1N∑A=1Nf(MA).We will now proceed to explicitly apply Wick's theorem to calculate the expectation values of permutation invariant functions labelled by graphs for the case of quadratic functions (2-edge graphs), cubic (3-edge graphs) and quartic functions (4-edge graphs). We will leave the comparison of the results of this 13-parameter Gaussian model to linguistic data for the future.3Graph basis invariants in terms of rep theory parametersIn the graph theoretic description of SD invariants constructed from Mij, nodes in the graph correspond to indices, M corresponds to directed edges. At linear order we have the one-node invariant ∑iMii and the two-node invariant ∑i,jMij. At quadratic order in M we have up to three nodes. In this section, we calculate the expectation values of the linear and quadratic invariants in the Gaussian model defined in Section 2. The quadratic expectation values show non-trivial mixing between the different Mij, unlike the 5-parameter model, where the Mii are decoupled from the off-diagonal elements and from each other. In that simple model, Mij only mixes with Mji. Here the mixings are more non-trivial, but controlled by SD representation theory.The quantity F(i,j) defined in (2.21)(3.1)F(i,j)=∑aCa,iCa,j=(δij−1D) will play an important role in the following. Its meaning is that it is the projector for the hook representation in the natural representation. Deriving expressions for expectation values of permutation invariant polynomial functions of the matrix variable M amounts to doing appropriate sums of products of F factors, with the arguments of these F factors being related to each other according to the nature of the polynomial under consideration. In terms of the variablesμ˜α=∑b=12(ΛV0−1)αβμβ defined in Section 2, repeated here for convenience,(3.2)〈SV0;1〉=μ˜1〈SV0;2〉=μ˜2. Using the expansion (2.59) of the matrix variables in terms of the rep-theoretic S variables and the 1-point functions of these in (2.74), (2.75) and (2.76), we have the 1-point function for the matrix variables(3.3)〈Mij〉=μ˜1D+μ˜2D−1F(i,j).Using (A.4), (A.5) along with the expansion of M in terms of S variables (equation 2.59), and the two-point functions of the S-variables, we have(3.4)〈MijMkl〉=〈MijMkl〉conn+〈Mij〉〈Mkl〉 where(3.5)〈MijMkl〉conn=1D2〈SV0;1SV0;1〉conn+1D−1∑a1,a2=1D−1Ca1,iCa1,jCa2,kCa2,l〈SV0;2SV0;2〉conn+1DD−1∑a=1D−1〈SV0;1SV0;2〉connCa,kCa,l+1DD−1∑a=1D−1〈SV0;2SV0;1〉connCa,iCa,j+1D∑a1,a2=1D−1Ca1,jCa2,l〈Sa1H;1Sa2H;1〉conn+1D∑a1,a2=1D−1Ca1,iCa2,k〈Sa1H;2Sa2H;2〉conn+∑a1,b1,a2,b2=1D−1Ca1,iCb1,jCa2,kCb2,lCa1,b1;c1HH→HCa2,b2;c2HH→H〈Sc1H;3Sc2H;3〉conn+1D∑a1,a2=1D−1Ca1,jCa2,k〈Sa1H;1Sa2H;2〉conn+1D∑a1,a2=1D−1Ca1,iCa2,l〈Sa1H;2Sa2H;1〉conn+1D∑a1=1D−1∑a2,b2=1D−1Ca1,jCa2,kCb2,lCa2,b2;c2HH→H〈Sa1H;1Sc2H;3〉conn+1D∑a1,b1=1D−1∑a2=1D−1Ca1,iCb1,jCa1,b1;c1HH→HCa2,l〈Sc1H;3Sa2H;1〉conn+1D∑a1=1D−1∑a2,b2=1D−1Ca1,iCa2,kCb2,lCa2,b2;c2HH→H〈Sa1H;2Sc2H;3〉conn+1D∑a1,b1=1D−1∑a2=1D−1Ca1,iCb1,jCa1,b1;c1HH→HCa2,k〈Sc1H;3Sa2H;2〉conn+∑a1,b1=1D−1∑a2,b2=1D−1Ca1,iCb1,jCa1,b1;c1HH→V2Ca2,kCb2,lCa2,b2;c2HH→V2〈Sc1V2Sc2V2〉conn+∑a1,b1=1D−1∑a2,b2=1D−1Ca1,iCb1,jCa1,b1;c1HH→V3Ca2,kCb2,lCa2,b2;c2HH→V3〈Sc1V3Sc2V3〉conn.All the terms can be expressed in terms of the F-function defined in (3.1)(3.6)〈MijMkl〉conn=1D2(ΛV0−1)11+(ΛV0−1)22(D−1)F(i,j)F(k,l)+(ΛV0−1)12DD−1(F(k,l)+F(i,j))+(ΛH−1)11DF(j,l)+(ΛH−1)22DF(i,k)+D(ΛH−1)33(D−2)∑p,q=1DF(i,p)F(j,p)F(k,q)F(l,q)F(p,q)+(ΛH−1)12D(F(j,k)+F(i,l))+(ΛH−1)13D−2(∑p=1DF(j,p)F(k,p)F(l,p)+F(i,p)F(j,p)F(l,p))+(ΛH−1)23D−2(∑p=1DF(i,p)F(k,p)F(l,p)+F(i,p)F(j,p)F(k,p))+(ΛV2−1)(12F(i,k)F(j,l)+12F(i,l)F(j,k)−DD−2∑p,q=1DF(i,p)F(j,p)F(k,q)F(l,q)F(p,q)−1(D−1)F(i,j)F(k,l))+(ΛV3−1)2(F(i,k)F(j,l)−F(i,l)F(j,k)). We will refer to the terms depending on ΛV0 as V0-channel contributions to the 2-point functions, those on ΛH as VH-channel (or H-channel) contributions, those on ΛV2 as V2-channel and those on ΛV3 as V3-channel contributions. It will be convenient to denote these different channel contributions as 〈MijMkl〉V where V∈{V0,VH,V2,V3}, so that we have(3.7)〈MijMkl〉conn=〈MijMkl〉connV0+〈MijMkl〉connVH+〈MijMkl〉connV2+〈MijMkl〉connV3.In arriving at the expressions for the last two terms in (3.6), we used the fact that these terms in (3.5) can be expressed as(3.8)ΛV2−1(ei⊗ej,PVD⊗VD→V2(ek⊗el))+ΛV3−1(ei⊗ej,PVD⊗VD→V3(ek⊗el)). Here PVD⊗VD→V2 is the projector from the tensor product of natural reps. to V2, the irrep of SD associated with Young diagram [D−2,2]. Similarly for V3=[D−2,1,1]. Now it is useful to observe that V3 is just the anti-symmetric part of VH⊗VH. The symmetric part decomposes as V0⊕VH⊕V2. The projector for V0 is(3.9)(PHH→V0)a1b1;a2b2=1D−1δa1a2δb1b2. For VH it is(3.10)(PHH→H)a1b1;a2b2=∑c=1D−1Ca1,b1;cH,H→HCa2,b2;cH,H→H=DD−2∑c=1D−1Ca1,b1,cCa2,b2,c. The factor DD−2 is explained in Appendix C.3.1Calculation of ∑i,j〈MijMij〉Following (3.7) the expectation value 〈MijMij〉 can be written as a sum over V-channel contributions, where V ranges over the four irreps.3.1.1Contributions from V2,V3From (3.8) we find(3.11)∑i,j〈MijMij〉connV2=(ΛV2)−1trVD⊗VD(PVD⊗VD→V2)=(DimV2)(ΛV2)−1=D(D−3)2(ΛV2)−1. The projector has eigenvalue 1 on the subspace transforming in the irrep V2 and zero elsewhere, hence the (DimV2). Similarly(3.12)∑i,j〈MijMij〉connV3=(DimV3)(ΛV3)−1=(D−1)(D−2)2(ΛV3)−1.Since V2,V3 appear inside the VH⊗VH subspace of Vnat⊗Vnat, we can also write the trace in VH⊗VH and express this in terms of irreducible characters(3.13)trVH⊗VHPV2=(DimV2)D!∑σ∈SDχV2(σ)χH(σ)χH(σ)=(DimV2). In the last line we have used the fact that the Kronecker coefficient for VH⊗VH→V2 is 1.3.1.2Contribution from VH channelThe (ΛH−1)11 contribution is(3.14)(DimVH)(ΛH−1)11=(D−1)(ΛH−1)11 which can be obtained easily from (3.5) or (3.6). Similarly, the (ΛH−1)12 contribution is zero. And the (ΛH−1)22 contribution is(3.15)(DimVH)(ΛH−1)22=(D−1)(ΛH−1)22.From either (3.5) or (3.6) we easily conclude that the (ΛH)13−1,(ΛH)23−1 contributions vanish. Starting from (3.5), we make repeated use of (2.19).The (ΛH−1)33 contribution is(3.16)(ΛH−1)33trVH⊗VH(PHH→H)=(ΛH−1)33(DimVH)=(ΛH−1)33(D−1). We used the second equation in (2.18).3.1.3Contribution from V0 channelThis is(3.17)∑i,j〈MijMij〉connV0=(ΛV0−1)11+1DimVH(ΛV0−1)22DimVH=(ΛV0−1)11+(ΛV0−1)22.3.1.4Summing all channels(3.18)∑i,j〈MijMij〉conn=(ΛV0−1)11+(ΛV0−1)22+(D−1)(ΛH−1)22+(D−1)(ΛH−1)33+(D−1)(ΛH−1)11+D(D−3)2(ΛV2)−1+(D−1)(D−2)2(ΛV3)−1.The disconnected piece is(3.19)∑i,j〈Mij〉〈Mij〉=∑i,j(μ˜1D+μ˜2D−1F(i,j))2=μ˜12+μ˜22, and using (3.4)(3.20)∑i,j〈MijMij〉=μ˜12+μ˜22+∑i,j〈MijMij〉conn=μ˜12+μ˜22+(ΛV0−1)11+(ΛV0−1)22+(D−1)(ΛH−1)22+(D−1)(ΛH−1)33+(D−1)(ΛH−1)11+D(D−3)2(ΛV2)−1+(D−1)(D−2)2(ΛV3)−1.3.2Calculation of ∑i,j〈MijMji〉As in (3.7) the expectation value 〈MijMij〉 can be written as a sum over V-channel contributions, where V ranges over the four irreps.3.2.1Contribution from multiplicity 1 channels V2,V3The (ΛV2−1) contribution is(3.21)(ΛV2−1)trVH⊗VH(PV2τ)=(ΛV2−1)dV2=(ΛV2−1)D(D−3)2. τ is the swop which acts on the two factors of VH. We have used the fact that V2 appears in the symmetric part of VH⊗VH.The (ΛV3−1) contribution is(3.22)−(ΛV3−1)(D−1)(D−2)2. We use the fact that V3 is the antisymmetric part of VH⊗VH.3.2.2Contribution from VH channelThe (ΛH−1)11 contribution From (3.5), we have(3.23)1D∑a1,a2∑i,jCa1,jCa2,i(ΛH−1)11δa1a2=1D(ΛH−1)11∑aCa,iCa,j=0 using (2.19). Similarly the (ΛH−1)22 contribution is zero.The (ΛH−1)12 contribution is(3.24)2(D−1)D∑a,jCa,jCa,j(ΛH−1)12=2(D−1)D(ΛH−1)12.The (ΛH−1)33 contribution is(3.25)∑i,j∑a,b,c,d,e(ΛH−1)33Ca,iCb,jCc,jCd,iCa,b;eHH→HCc,d;eHH→H=(ΛH−1)33∑a,b,c,d,eCa,b;eHH→HCb,a;eHH→H=(ΛH−1)33trH⊗HPH⊗H→Hs=(ΛH−1)33trH⊗HPH⊗H→H(1+s2−1−τ2)=(ΛH−1)33dH=(D−1)(ΛH−1)33.In the penultimate line, we have introduced the swop s which exchanges the two factors in H⊗H. We know that H appears in the symmetric part of H⊗H, so the swop leaves it invariant.Use of the equation (2.19) shows that the (ΛH−1)13,(ΛH−1)23 dependent terms vanish.Collecting all the VH-channel contributions, we have(3.26)∑i,j〈MijMji〉connVH=2(D−1)D(ΛH−1)12+(D−1)(ΛH−1)33.3.2.3Contribution from V0 channelThe first term from (3.5) is(3.27)(ΛV0−1)11. The second term is(3.28)(ΛV0−1)22dH∑i,j∑a1,a2Ca1,iCa1,jCa2,jCa2,i=(ΛV0−1)22D−1∑a1,a2δa1a2δa1a2=(ΛV0−1)22. The third term is(3.29)1DD−1∑i,j∑aCa,jCa,i(ΛV0−1)12, which vanishes using (2.19). The last term vanishes for the same reason.So collecting the V0-channel contributions to ∑i,j〈MijMji〉conn, we have(3.30)∑i,j〈MijMji〉conn=(ΛV0−1)11+(ΛV0−1)22.3.2.4Summing all channels(3.31)∑i,j〈MijMji〉conn=(ΛV2−1)(D)(D−3)2−(ΛV3−1)(D−1)(D−2)2+2(D−1)D(ΛH−1)12+(D−1)(ΛH−1)33+(ΛV0−1)11+(ΛV0−1)22.Since F(i,j)=F(j,i), the disconnected piece is the same in (3.19)(3.32)∑i,j〈Mij〉〈Mji〉=∑i,j〈Mij〉〈Mij〉=μ˜12+μ˜22, and ∑i,j〈MijMij〉 in terms of the μ,Λ parameters of the Gaussian model is the sum of the expressions in (3.31) and (3.32).3.3Calculation of ∑i,j〈MiiMij〉An important observation here is that the sum over j projects the representation VD to the trivial irrep V0, which follows from the formula for C0i in (2.17). This means that when we expand Mii and Mij into S variables as in the first line of (2.54), we only need to keep the term SH0 or S00 from the expansion of Mij.3.3.1Contribution from V2,V3 channelsFrom the above observation, and since V2,V3 appear only in 〈SHHSHH〉, we immediately see that(3.33)∑i,j〈MiiMij〉connV2=∑i,j〈MiiMij〉connV3=0.3.3.2Contribution from VH channelFrom the above observation, the only non-zero contributions in the VH channel come from 〈SHH→HSH0〉, 〈SH0SH0〉 and 〈S0HSH0〉. These are(3.34)1D∑a1,a2,iCa1,iCa2,i(ΛH−1)12δa1a2=(D−1)D(ΛH−1)12(3.35)1D∑a1,a2,iCa1,iCa2,i(ΛH−1)22δa1,a2=(D−1)D(ΛH−1)22(3.36)1D∑a1,b1,a2Ca1,iCb1,iCa1,b1;c1H,H→HCa2,iδc1a2(ΛH−1)32=1D−2∑a1,b1,a2Ca1,iCb1,iCa1,b1,c1Ca2,iδc1a2(ΛH−1)32=1D(D−1)(D−2)(ΛH−1)23. Note that ΛH is a symmetric 3×3 matrix and ΛH−1)23=(ΛH−1)32. These add up to(3.37)〈MiiMij〉connVH=(D−1)D(ΛH−1)12+(D−1)D(ΛH−1)22+1D(D−1)(D−2)(ΛH−1)23.3.3.3Contribution from V0 channelThe non-zero contributions come from 〈S00S00〉 and 〈SHH→0S00〉. They are(3.38)∑i,j〈MiiMij〉connV0=1D2∑i,j(ΛV0−1)11+1DdH∑i,j∑a(ΛV0−1)12Ca,iCa,i=(ΛV0−1)11+1D−1∑i,a(ΛV0−1)Ca,iCa,i=(ΛV0−1)11+dH(ΛV0−1)12=(ΛV0−1)11+(D−1)(ΛV0−1)12.3.3.4Summing all channels(3.39)∑i,j〈MiiMij〉conn=(ΛV0−1)11+(D−1)(ΛV0−1)12+(D−1)D(ΛH−1)12+(D−1)D(ΛH−1)22+1D(D−1)(D−2)(ΛH−1)23.The disconnected piece is(3.40)∑i,j〈Mii〉〈Mij〉=(μ˜1D+μ˜2D−1F(i,i))(μ˜1D+μ˜2D−1F(i,j))=μ˜12+μ˜1μ˜2D−1∑i,j(1−1D)=μ˜12+μ˜1μ˜2D(D−1) so we have(3.41)∑i,j〈MiiMij〉=∑i,j〈MiiMij〉conn+μ˜12+μ˜1μ˜2DD−1, with the first term given by (3.39)3.4Calculation of ∑i,j〈MiiMji〉We can write down the answer from inspection of (3.39)(3.42)∑i,j〈MiiMji〉conn=(ΛV0−1)11+(D−1)(ΛV0−1)12+(D−1)D(ΛH−1)12+(D−1)D(ΛH−1)11+1D(D−1)(D−2)(ΛH−1)13.The reasoning is as follows. The sum over j projects to V0. This means that the only non-zero contributions are, from the V0 channel,(3.43)〈S00S00〉conn〈SHH→0S00〉conn and from the VH channel(3.44)〈S0HS0H〉conn〈SH0S0H〉conn〈SHH→HS0H〉conn. This identifies the contributing entries of ΛV0−1,ΛH−1 using the indexing in (2.51) and (2.52). Given the similarity between the expectation value in section 3.3, we have contributions of the same form, up to taking care of the right indices on ΛV0−1,ΛH−1.Given the symmetry of F(i,j) under exchange of i,j, the disconnected piece is the same as above(3.45)∑i,j〈MiiMji〉=(ΛV0−1)11+(D−1)(ΛV0−1)12+(D−1)D(ΛH−1)12+(D−1)D(ΛH−1)11+1D(D−1)(D−2)(ΛH−1)13+μ˜12+μ˜1μ˜2D(D−1).3.5Calculation of ∑i,j,k〈MijMik〉The sums over j,k project to V0. The non-vanishing contributions are 〈S00S00〉conn and 〈SH0SH0〉conn. They add up to(3.46)∑i,j,k〈MijMik〉conn=D(ΛV0−1)11+D(D−1)(ΛH−1)22. The disconnected part is(3.47)∑i,j,k〈Mij〉〈Mik〉=μ˜12D leading to(3.48)∑i,j,k〈MijMik〉=D(ΛV0−1)11+D(D−1)(ΛH−1)22+μ˜12D.3.6Calculation of ∑i,j,k〈MijMkj〉Now we are projecting to V0 on the first index of both M's. This means that the contributing terms are 〈S00S00〉 and 〈S0HS0H〉.Repeat the same steps as above in (3.46) to get(3.49)∑i,j,k〈MijMkj〉con=D(ΛV0−1)11+D(D−1)(ΛH−1)11. The only difference is that we are picking up the (1,1) matrix element of (ΛH−1) instead of the (2,2) element, since we defined S0H=S{V0;1} and SH0=S{V0;2}.Adding the disconnected piece, which is the same as (3.47), we have(3.50)∑i,j,k〈MijMkj〉=D(ΛV0−1)11+D(D−1)(ΛH−1)11+μ˜12D.3.7Calculation of ∑i,j,k〈MijMjk〉We are now projecting to V0 on first index of one of the matrices and second index of the other. Hence the contributing terms are 〈S00S00〉 and 〈S0HSH0〉. The result is(3.51)∑i,j,k〈MijMjk〉conn=D(ΛV0−1)11+D(D−1)(ΛH−1)12, and(3.52)∑i,j,k〈MijMjk〉=D(ΛV0−1)11+D(D−1)(ΛH−1)12+μ˜12D.3.8Calculation of ∑i,j,k,l〈MijMkl〉Here we project to V0 on all four indices, so(3.53)∑i,j,k,l〈MijMkl〉conn=D2〈S00S00〉=D2(ΛV0−1)11. Adding the disconnected piece we have(3.54)∑i,j,k,l〈MijMkl〉=D2(Λ−1V0)11+μ˜12D2.3.9Calculation of ∑i〈Mii2〉3.9.1The V0 channelThe contribution from the V0 channel is given by(3.55)∑i〈Mii2〉connV0=1D2∑i(ΛV0−1)11+1(D−1)∑iF(i,i)2(ΛV0−1)22+2DD−1∑iF(i,i)(ΛV0−1)12=1D(ΛV0−1)11+(D−1)D(ΛV0−1)22+2D−1D(ΛV0−1)12.3.9.2The VH channelIt is convenient to use (3.6) to arrive at(3.56)∑i〈Mii2〉connVH=D−1D(ΛH−1)11+(ΛH−1)22(D−1)D+D−1(D−1)(D−2)(ΛH−1)33+2(D−1)D(ΛH−1)12+2(ΛH−1)13D(D−1)(D−2)+2(ΛH−1)23D(D−1)(D−2). Useful equations in arriving at the above are the sums(3.57)∑i,p,qF(i,p)F(i,p)F(i,q)F(i,q)F(p,q)=(D−2)2(D−1)D2∑i,p(F(i,p))3=(D−1)(D−2)D, which can be obtained by hand or with the help of Mathematica. In the latter case, it is occasionally easier to evaluate for a range of integer D and fit to a form Polynomial(D)/Dsomepower.3.9.3The V2,V3 channelsNow calculate the HH→V2 and HH→V3 channel.(3.58)∑i〈Mii2〉connV2=∑i∑a,b,c,dCa,iCb,iCc,iCd,iCab;eHH;V2Ccd;fHH;V2〈SeHH→V2SfHH→V2〉conn=(ΛV2−1)∑i,j∑a,b,c,d,eCa,iCb,iδijCc,jCd,jCa,b;eHH;V2Cc,d;eHH;V2=(ΛV2−1)∑a,b,c,d,eCa,iCb,i(Ce,iCe,j+1D)Cc,jCd,jCa,b;eHH;V2CHH;V2c,d;e=D−2D(ΛV2−1)trH⊗H(PHPV2)+1D(ΛV2−1)∑a,bCa,a;eHH→V2CHH→V2b,b;e=D−2D(ΛV2−1)trH⊗H(PH,H→HPH,H→V2)+1D(ΛV2−1)∑a,bCa,a;eHH→V2Cb,b;eHH→V2=0+(ΛV2−1)D−2∑a,b,i,jCa,iCa,iCe,iCb,jCb,jCe,j=(ΛV2−1)D−2∑i,jF(i,j)F(i,i)F(j,j)=(ΛV2−1)(D−1)2D2(D−2)∑i,jF(i,j)=0. We used the fact that the projectors for H,V2 are orthogonal.Similarly, the contribution from V3 is zero. Another to arrive at the same answer is to recognise that V3 is the antisymmetric part, so(3.59)Pab;cdH,H→V3=12(δacδbd−δadδbc).3.9.4Summing the channels(3.60)∑i〈Mii2〉conn=1D(ΛV0−1)11+(D−1)D(ΛV0−1)22+2D−1D(ΛV0−1)12+D−1D(ΛH−1)11+(ΛH−1)22(D−1)D+D−1(D−1)(D−2)(ΛH−1)33+2(D−1)D(ΛH−1)12+2(ΛH−1)13D(D−1)(D−2)+2(ΛH−1)23D(D−1)(D−2).The disconnected part is(3.61)∑i〈Mii〉〈Mii〉=∑i(μ˜1D+μ˜2D−1F(i,i))2=μ˜12D+2μ˜1μ˜2D−1D+μ˜22(D−1)D.3.10Calculation of ∑i,j〈MiiMjj〉Since ∑iMii and ∑jMjj are SD invariant, we only have contributions from the V0 channel. Use the first four terms of (3.5) to get(3.62)∑i,j〈MiiMjj〉conn=D(ΛV0−1)11+dH(ΛV0−1)22+2dHD(Λ0−1)12. Using(3.63)∑i(μ˜1D+μ˜2D−1F(i,i))=μ˜1+μ˜2D−1 the disconnected part is(3.64)∑i,j〈Mii〉〈Mjj〉=μ˜12+2μ˜1μ˜2D−1+μ˜22(D−1) so that(3.65)∑i,j〈MiiMjj〉=∑i,j〈MiiMjj〉conn+μ˜12+2μ˜1μ˜2D−1+μ˜22(D−1).3.11Calculation of ∑i,j,k〈MiiMjk〉Here we get contributions from 〈SHH→0S00〉conn and 〈S00S00〉conn. Adding these up from (3.5)(3.66)∑i,j,k〈MiiMjk〉conn=D(ΛV0−1)11+D−1D(ΛV0−1)12.The disconnected part is(3.67)∑i,j,k〈Mii〉〈Mjk〉=(μ˜1+μ˜2D−1)Dμ˜1 hence(3.68)∑i,j,k〈MiiMjk〉=D(ΛV0−1)11+D−1D(ΛV0−1)12+(μ˜1+μ˜2D−1)Dμ˜1.3.12Summary of results for quadratic expectation values in a large D limitIt is interesting to collect the results for the connected quadratic expectation values and consider the large D limit. Let us assume that all the ΛV0,ΛH,ΛV2,ΛV3 scale in the same way as D→∞ and consider the sums normalized by the appropriate factor of D1D2∑i,j〈MijMij〉conn=12((ΛV2)−1+(ΛV3)−1)1D2∑i,j〈MijMji〉conn=12((ΛV2)−1−(ΛV3)−1)1D2∑i,j〈MiiMij〉conn=1D3/2(ΛV0−1)12+1D3/2(ΛH−1)231D2∑i,j〈MiiMji〉conn=1D3/2(ΛV0−1)12+1D3/2(ΛH−1)131D3∑i,j,k〈MijMik〉conn=1D(ΛH−1)221D3∑i,j,k〈MijMkj〉conn=1D(ΛH−1)111D3∑i,j,k〈MijMjk〉conn=1D(ΛH−1)121D4∑i,j,k,l〈MijMkl〉conn=1D2(ΛV0−1)111D∑i〈Mii2〉conn=(ΛH−1)331D2∑i,j〈MiiMjj〉conn=1D(ΛV0−1)11+1D(ΛV0−1)221D3∑i,j,k〈MiiMjk〉conn=1D2(ΛV0−1)11. The dominant expectation values in this limit are the first, second and ninth. These are the quadratic expressions which enter the simplified 5-parameter model considered in [8] (see Equation (1.1)). It will be interesting to systematically explore the different large D scalings of the parameters in real world data, e.g. the computational linguistics setting of [8] or in any other situation where permutation invariant matrix Gaussian matrix distributions can be argued to be appropriate.4A selection of cubic expectation valuesIn this section we use Wick's theorem from Appendix A to express expectation values of cubic functions of matrix variables in terms of linear and quadratic expectation values. The permutation invariance condition requires sums of indices over the range {1,⋯,D}. This leads to non-trivial sums of products of the natural-to-hook projector F(i,j). The invariants at cubic order are 52 in number (Appendix B of [8]) and correspond to graphs with up to 6 nodes.4.11-node case ∑i〈Mii3〉Using (A.6), we have(4.1)∑i〈Mii3〉=3∑i〈Mii2〉conn〈Mii〉+〈Mii〉3. Specialising (3.3)(4.2)〈Mii〉=1Dμ˜1+(D−1)Dμ˜2. Since this is independent of i, we can use (3.60) to get(4.3)∑i〈Mii3〉=(1Dμ˜1+(D−1)D)μ˜2)×(1D(ΛV0−1)11+(D−1)D(ΛV0−1)22+2D−1D(ΛV0−1)12+D−1D(ΛH−1)11+(ΛH−1)22(D−1)D+D−1(D−1)(D−2)(ΛH−1)33+2(D−1)D(ΛH−1)12+2(ΛH−1)13D(D−1)(D−2)+2(ΛH−1)23D(D−1)(D−2))+1D2(μ˜1+(D−1)μ˜2)3.4.2A 2-node case ∑i,j〈Mij3〉Using (A.6) we have(4.4)〈Mij3〉=∑i,j3〈Mij2〉conn〈Mij〉+∑i,j〈Mij〉3. Calculating this requires doing a few sums, which can be done by hand or with Mathematica (the function KoneckerDelta is handy).(4.5)∑i,jF(i,j)=0∑i,j(F(i,j))2=(D−1)∑i,j(F(i,j))3=D−1(D−1)(D−2). Using (3.3), we find for the second term in (4.4)(4.6)∑i,j〈Mij〉3=μ˜13D+3Dμ˜1μ˜22+(D−2)DD−1μ˜23. For the first term on the RHS of (4.4)(4.7)∑i,j〈Mij〉conn〈Mij〉=3μ˜1D∑i,j〈Mij2〉conn+3μ˜2D−1∑i,j〈Mij2〉connF(i,j). The first term in (4.7) can be expressed as a function of the parameters of the Gaussian model using (3.18). The second term is calculated by specialising the fundamental quadratic moments (3.6) and doing the resulting sums over the F-factors. Consider the V0 contributions to the second term above. The term proportional to (ΛV0−1)11 vanishes due to the first of (4.5). The (22) contribution, using the third of (4.5) is(4.8)3μ˜2D−1×D−2D(ΛV0−1)22=3μ˜2(ΛV0−1)22(D−2)DD−1. The (12) contributions, using the second of (4.5) is(4.9)3μ˜2D−1×2D−1D(ΛV0−1)12=6μ˜2(ΛV0−1)12D. Now consider the VH contribution to the second term in (4.7). The (ΛH−1)11 term is(4.10)1D(ΛH−1)11∑i,jF(i,j)F(j,j)=0. The (ΛH−1)22 contribution is similarly zero. The (ΛH−1)33 contribution is(4.11)3μ˜2D−1×D(ΛH−1)33(D−2)×∑i,j,p,qF(i,p)F(i,q)F(j,p)F(j,q)F(i,j)F(p,q)=3μ˜2(ΛH−1)33D−1(D−3)D. The sum of products of six F's is readily done with Mathematica to give D−2(D−1)(D−2)(D−3).Contributions from the (1,2) matrix element of symmetric matrix ΛH give(4.12)6μ˜2(D−1)D(ΛH−1)12. This uses the second of (4.5). From (1,3) and (3,1) we have(4.13)3μ˜2D−1×ΛH−1D−2×2∑i,j,pF(i,p)(F(j,p))2F(i,j)=3μ˜2D−1×ΛH−1D−2×2D−1(D−1)(D−2)=6μ˜2D(ΛH−1)13(D−1)(D−2). From (2,3) and (3,2), we have(4.14)6μ˜2D(D−1)(D−2)(ΛH−1)23.Now consider the contribution from V2. It is convenient to use (3.5)(4.15)(ΛV2−1)∑i,j∑a1,b1,c,a2,b2,dCa1iCb1jCa1b1cH,H→V2Ca2iCb2jCa2,b2,cH,H→V2CdiCdj=(ΛV2−1)∑i,j∑a1,b1,c,a2,b2,dCa1iCa2iCdiCb1jCb2jCdj(PH,H→V2)a1b1;a2b2=(ΛV2−1)∑a1,b1,c,a2,b2,dCa1a2dCb1b2d(PH,H→V2)a1b1;a2b2=(ΛV2−1)D−2D∑a1,a2,b1,b2(PH,H→H)a1,a2;b1,b2(PH,H→V2)a1b1;a2b2. In the last line, we have used the second equation in C.10 which gives the relation between the invariant Cabc in Vnat⊗3 and the normalized Clebsch-Gordan coefficients CabcHH→H, and the formula for the projector in terms of the Clebsch-Gordan coefficients.The symmetric part of VH⊗VH, i.e. the subspace invariant under the swop of the two factors, decomposes into irreducible representations of the diagonal SD action as V0⊕VH⊕V2. This means that(4.16)PH,H→V2=(1−PH,H→H−PH,H→V0)(1+τ)2PH,H→H(1+τ)2=PH,H→H. This means that(4.17)(PH,H→V2)a1b1;a2b2=12(δa1a2δb1b2+δa1b2δa2b1)−Pa1,b1;a2,b2H,H→H−PH,H→V0a1,b1;a2,b2. A useful fact following from (2.21) and (2.19) is(4.18)∑aCaab=0. When the expression (4.17) is substituted in (4.15) the first term on the RHS of (4.17) does not contribute because of (4.18). The second term gives(4.19)(ΛV2−1)D−22D∑a1,a2,b1,b2(PH,H→H)a1,a2;a2,a1=(ΛV2−1)(D−1)(D−2)2D. The third term gives(4.20)−(ΛV2−1)D−2D∑a1,a2,b1,b2(PH,H→H)a1,a2;b1,b2(PH,H→H)a1b1;a2b2=−(ΛV2−1)D(D−2∑a1,a2,b1,b2,c1,c2Ca1,a2,c1Cb1,b2,c1Ca1,b1,c2Ca2,b2,c2=−(ΛV2−1)D(D−2)∑a1,a2,b1,b2,c1,c2∑i,j,p,qF(i,p)F(i,q)F(i,j)F(j,p)F(j,q)F(p,q)=−(ΛV2−1)D(D−2)(D−1)(D−2)(D−3)D2=−(ΛV2−1)(D−1)(D−3)D. The fourth term gives(4.21)−(ΛV2−1)D−2D∑a1,a2,b1,b2(PH,H→H)a1,a2;b1,b2(PH,H→V0)a1,b1;a2,b2−(ΛV2−1)D−2D1(D−1)∑a1,a2,b1,b2(PH,H→H)a1,a2;b1,b2δa1b1δa2b2=−(ΛV2−1)D−2D. Collecting terms from (4.19), (4.20) and (4.21) we get(4.22)(ΛV2−1)(D2−3D+3)2D. Multiplying the factor 3μ˜2D−1 from (4.7) to get a contribution to ∑i,j〈Mij3〉, we get(4.23)3μ˜2D−1(ΛV2−1)(D2−3D+3)2D=μ˜2ΛV2−13(D2−3D+3)2DD−1.The contribution from V3 is(4.24)3μ˜2D−1(ΛV3−1)D−2D∑a1,a2,b1,b2(PH,H→H)a1,a2;b1,b2(PH,H→V3)a1b1;a2b2. Now use the fact that(4.25)(PH,H→V3)a1b1;a2b2=12(δa1a2δb1b2−δa1b2δa2b1) along with(4.26)∑a,b(PH,H→H)aa;bb=0∑a1,a2(PH,H→H)a1a2;a2a1=∑a1,a2(PH,H→H)a1a2;a1a2=(D−1) to find(4.27)−(ΛV3−1)3μ˜2D−1(D−2)D(D−1)=−3μ˜2(ΛV3−1)(D−2)D−1D.Collecting all the contributions we have(4.28)∑i,j〈Mij3〉=μ˜13D+3Dμ˜1μ˜22+(D−2)DD−1μ˜23+3μ˜1D((ΛV0−1)11+(ΛV0−1)22+(D−1)(ΛH−1)22+(D−1)(ΛH−1)33+(D−1)(ΛH−1)11+D(D−3)2(ΛV2)−1+(D−1)(D−2)2(ΛV3)−1)+3μ˜2(ΛV0−1)22D−2DD−1+6μ˜2(ΛV0−1)12D+3μ˜2(ΛH−1)33D−1(D−3)D+6μ˜2(ΛH−1)12(D−1)D+6μ˜2(ΛH−1)13(D−1)(D−2)D+6μ˜2(ΛH−1)13(D−1)(D−2)D+μ˜2ΛV2−13(D2−3D+3)2DD−1−3μ˜2(ΛV3−1)(D−2)D−1D.4.3A 3-node case ∑i,j,k〈MijMjkMki〉Using Wick's theorem (A.6)(4.29)∑i,j,k〈MijMjkMki〉=∑i,j,k〈Mij〉〈Mjk〉〈Mki〉+〈MijMjk〉〈Mki〉+〈MijMki〉〈Mjk〉+〈MjkMki〉〈Mij〉.The first term is(4.30)∑i,j,k(μ˜1D+μ˜2D−1F(i,j))(μ˜1D+μ˜2D−1F(j,k))(μ˜1D+μ˜2D−1F(k,i)). Using(4.31)∑jF(i,j)F(j,k)=F(i,k) along with the first and second of (4.5) we can show that(4.32)∑i,j,k〈Mij〉〈Mjk〉〈Mki〉=μ˜13+μ˜23D−1. Consider the remaining three terms. Focus on the first of these:(4.33)∑i,j,k〈MijMjk〉conn〈Mki〉=∑i,j,k〈MijMjk〉conn(μ˜1D+μ˜2D−1F(k,i))=μ˜1D∑i,j,k〈MijMjk〉conn+μ˜2D−1∑i,j,k〈MijMjk〉F(k,i). We already know the first term from (3.51). So let us consider the second. An easy calculation using (3.5) (or equivalently using (3.6)) shows that the contribution from the V0 channel is(4.34)μ˜2D−1(ΛV0−1)22. From the VH channel, the contributions are(4.35)μ˜2D−1(ΛH−1)33+μ˜2DD−1(ΛH−1)12. From the V2 channel, we get(4.36)μ˜2ΛV2−1D(D−3)2D−1−μ˜2(ΛV3−1)D−2D−12.Collecting terms(4.37)∑i,j,k〈MijMjk〉conn〈Mki〉=μ˜1((ΛV0−1)11+(D−1)(ΛH−1)12)+μ˜2D−1(Λ0−1)22+μ˜2D−1(ΛH−1)33+μ˜2DD−1(ΛH−1)12+μ˜2ΛV2−1D(D−3)2D−1−μ˜2(ΛV3−1)(D−2)D−12.By relabelling indices, it is easy to see that(4.38)∑i,j,k〈MijMjk〉conn〈Mki〉=∑i,j,k〈MijMki〉conn〈Mjk〉=∑i,j,k〈MjkMki〉conn〈Mij〉Hence, we have(4.39)∑i,j,k〈MijMjkMki〉=μ˜13+μ˜23D−1+3μ˜1((ΛV0−1)11+(D−1)(ΛH−1)12)+3μ˜2D−1(ΛV0−1)22+3μ˜2D−1(ΛH−1)33+3μ˜2DD−1(Λ−1H)12+3μ˜2Λ2−1D(D−3)2D−1−3μ˜2(Λ3−1)(D−2)D−12.4.4A 6-node case ∑i1,⋯,i6〈Mi1i2Mi3i4Mi5i6〉The sums over i1,⋯,i6 project to the V0 representations. As a result, using (A.6), along with (2.59), we have(4.40)∑i1,⋯,i6〈Mi1i2Mi3i4Mi5i6〉=3D3∑i1,⋯,i6〈SV0;1SV0;1〉〈SV0;1〉+1D3∑i1,⋯,i6(〈SV0;1〉)3=3μ˜1D3(ΛV0−1)11+μ˜13D3.5A selection of quartic expectation valuesThe methods we have used to calculate the cubic expectation values, which were explained in detail above, extend straightforwardly to quartic expectation values. The first step is to use Wick's theorem (A.7). Then we use the formulae for quadratic and linear expectation values from Sections 2 and 3. In order to arrive at the final result as a function of D,μ˜1,μ˜2,ΛV0,ΛH,ΛV2,ΛV3 we have to do certain sums over products of the natural-to-Hook projector F(i,j). We will give some formulae below to illustrate these steps for the quartic case, without producing detailed formulae as in previous sections.5.1A 2-node quartic expectation value ∑i,j〈Mij4〉(5.1)∑i,j〈Mij4〉=∑i,j〈Mij〉4+6∑i,j〈Mij2〉conn〈Mij〉2+3∑i,j〈Mij2〉conn〈Mij2〉conn.The quadratic average is(5.2)〈Mi,j2〉conn=(ΛV0−1)11D2+(ΛV0−1)22(D−1)2F(i,j)2+2(ΛV0−1)12DD−1F(i,j)+(ΛH−1)11DF(j,j)+(ΛH−1)22DF(i,i)+(ΛH−1)33D−2∑p,qF(i,p)F(j,q))F(i,q)F(j,q)F(p,q)+2(ΛH−1)12DF(i,j)+2(ΛH−1)13D−2∑pF(i,p)F(j,p)F(j,p)+2(ΛH−1)23D−2∑pF(i,p)F(j,p)F(j,p)+(ΛV2)−1(12F(i,i)F(j,j)+12F(i,j)F(i,j)−DD−2∑p,qF(i,p)F(i,q)F(j,p)F(j,q)F(p,q))+(ΛV3−1)2(12F(i,i)F(j,j)−F(i,j)F(i,j)).Using this and(5.3)〈Mij〉=(μ˜1D+μ˜2D−1F(i,j)), we can work out the formula for ∑i,j〈Mij4〉 as a function of the 13 Gaussian model parameters. Mathematica would be handy in doing the sums over products of F(i,j) which arise.5.2A 5-node quartic expectation value ∑i,j,k,p,q〈MijMjkMkpMpq〉From (A.7), we have(5.4)∑i,j,k,p,q〈MijMjkMkpMpq〉=∑i,j,k,p,q〈Mij〉〈Mjk〉〈Mkp〉〈Mpq〉+∑〈MijMjk〉conn〈Mkp〉〈Mpq〉+∑〈MijMkp〉conn〈Mjk〉〈Mpq〉+∑〈MijMpq〉conn〈Mjk〉〈Mkp〉+∑〈MjkMkp〉conn〈Mij〉〈Mpq〉+∑〈MjkMpq〉conn〈Mij〉〈Mkp〉+∑〈MkpMpq〉conn〈Mij〉〈Mjk〉+∑〈MijMjk〉conn〈MkpMpq〉conn+∑〈MijMkp〉conn〈MjkMpq〉conn+∑〈MijMpq〉conn〈MjkMkp〉conn. All the summands on the RHS can be evaluated using (3.5) or (3.6) in terms of F(i,j). The sums can be done with the help of Mathematica to obtain expressions in terms of D,μ˜1,μ˜2,ΛV.6Summary and outlookWe have used the representation theory of symmetric groups SD in order to define a 13-parameter permutation invariant Gaussian matrix model, to compute the expectation values of all the graph-basis permutation invariant quadratic functions of the random matrix, and a selection of cubic and quartic invariants. In [8] analogous computations with a 5-parameter model were compared with matrix data constructed from a corpus of the English language. A natural direction is to extend that discussion of the English language, or indeed other languages, to the present 13-parameter model. Combining the experimental methods employed in [8] with machine learning methods such those used in [18], in the investigation of the 13-parameter model, would also be interesting to explore.As a theoretical extension of the present work, it will be useful to generalise the representation theoretic parametrisation of the Gaussian models to perturbations of the Gaussian model, where we add cubic and quartic terms to the Gaussian action. Identifying parameter spaces of these deformations which allow well-defined convergent partition functions and expectation values will be useful for eventual comparison to data. If we ignore the convergence constraints, the general perturbed model at cubic and quartic order has 348 parameters, since there are 52 cubic invariants and 296 quartic invariants (Appendix A of [8]). As in the Gaussian case, we can expect that representation theory methods will be useful in handling this more general problem. Further techniques involving partition algebras underlying the representation theory of tensor products of the natural representation will likely play a role (see e.g. [19] for recent work in these directions).It is worth noting that permutation invariant random matrix distributions have been approached from a different perspective, based on non-commutative probability theory [20–22]. The approach of the present paper and [8] is based on the connection between statistical physics and zero dimensional quantum field theory (QFT). It would seem that the approach of the present paper can complement the theory developed in these papers [20–23] by producing integral representations (Gaussians or perturbed Gaussians) of random matrix distributions having finite expectation values for permutation invariant polynomial functions of matrices. The results on the central limit theorem from the above references would be very interesting to interpret from the present QFT perspective.The computation of expectation values in Gaussian matrix models admits generalization to higher tensors. Indeed the motivating framework in computational linguistics discussed in [8] involves matrices as well as higher tensors in a natural way. Generalizations of the present work on representation theoretic parametrisation of Gaussian models and computation of graph-basis observables to the tensor case is an interesting avenue for future research.In this paper, we have focused on the explicit computation of permutation invariant correlators for general D. Some simplifications at large D were discussed in section 3.12. For traditional matrix models having U(D) (or SO(D)/Sp(D) symmetries), there is a rich geometry of two dimensional surfaces and maps in the large D expansions which allows these expansions of matrix quantum field theories to have deep connections to string theory [27,28]. It will be interesting to explore the possibility of two dimensional geometrical interpretations of the large D expansion in permutation invariant matrix models.AcknowledgementsThis research is supported by the STFC consolidated grant ST/P000754/1 “String Theory, Gauge Theory & Duality” and a Visiting Professorship at the University of the Witwatersrand, funded by a Simons Foundation grant 509116 to the Mandelstam Institute for Theoretical Physics. I thank the Galileo Galilei Institute for Theoretical Physics for hospitality and the Italian National Institute for Nuclear Physics INFN for partial support during the completion of this work. I also thank the organizers of the workshop on Matrix Models for non-commutative geometry and string theory in Vienna, the KEK theory group, Tsukuba, Japan and the Yukawa Institute for Theoretical Physics (YITP), Kyoto, Japan, for hospitality during the completion of this project. I am grateful for conversations on the subject of this paper to David Arrowsmith, Masayuki Asahara, Robert de Mello Koch, Yang Hui He, Christopher Hull, Satoshi Iso, Vishnu Jejjala, Dimitrios Kartsaklis, Mehrnoosh Sadrzadeh, Shotaro Shiba. I thank the anonymous referee for Nuclear Physics B for a detailed reading and suggestions for an improved presentation.Appendix AMulti-dimensional Gaussian integrals and Wick's theoremConsider the multi-variable integral with a Gaussian integrand(A.1)Z=∫dxexp(−12∑i,j=1NxiAijxj+∑isixi)=(2π)NdetAexp(12si(A−1)ijsj). x∈RN. A∈CN×N is a real symmetric positive definite matrix. s∈RN is an arbitrary complex vector (see for example [16], [17], Appendix A, Equations (8) and (9) of [25]). One can also consider A more generally to be complex with positive definite real part, but to keep a probabilistic interpretation we keep A real symmetric. Expectation values of functions f(x) are defined by(A.2)〈f(x)〉=1Z∫dxf(x)exp(−12∑i,j=1NxiAijxj+∑isixi). These expectation values can be calculated by taking derivatives with respect to si on both sides of (A.1). For the x variables(A.3)〈xi〉=∑j(A−1)ijsj=∑jsj(A−1)ji. Application of this equation, along with the formula for dM in terms of the representation theoretic S-variables (2.67) leads to (2.74), (2.76). For expectation values of quadratic monomials we have(A.4)〈xixj〉=(A−1)ij+〈xi〉〈xj〉. We define the connected part as(A.5)〈xixj〉conn≡〈xixj〉−〈xi〉〈xj〉=(A−1)ij. The expressions (2.77) and (2.78) follow from these.For cubic expressions(A.6)〈xixjxk〉=〈xixj〉conn〈xk〉+〈xixk〉conn〈xj〉+〈xjxk〉conn〈xi〉+〈xi〉〈xj〉〈xk〉. For quartic expressions(A.7)〈xixjxkxl〉=〈xixj〉conn〈xkxl〉conn+〈xixk〉conn〈xjxl〉conn+〈xixl〉conn〈xjxk〉conn+〈xixj〉conn〈xk〉〈xl〉+〈xixk〉conn〈xj〉〈xl〉+〈xixl〉conn〈xj〉〈xk〉+〈xi〉〈xj〉〈xk〉〈xl〉. These illustrate a general fact (known as Wick's theorem in the quantum field theory context and Isserlis' theorem in probability theory [26]) about Gaussian expectation values. Higher order expectation values can be expressed in terms of linear and quadratic expectation values. When applied to permutation invariant matrix models, we still have non-trivial sums left to do, after Wick's theorem has been applied. This is illustrated in the calculations of section 4 and section 5.Appendix BRep theory of VH and its tensor productsSome basics of rep theory of VH can be presented in a self-contained way, assuming only knowledge of linear algebra and index notation.Alternatively, we can observe that the matrices in VH are the same as Young's orthogonal basis. If we just follow the self-contained route, we define(B.1)DabH(σ)=(Ea,σEb)=∑i∑jCa,iCb,j(ei,ej)=∑i,jCa,iCb,j(ei,eσ−1(j))=∑i,jCa,iCb,jδi,σ−1(j)=∑iCa,iCb,σ(i).We have the orthogonality property:(B.2)DabH(σ−1)=∑iCa,iCb,σ−1(i)=∑iCa,σ(i)Cb,i=DbaH(σ).The homomorphism property(B.3)DabH(σ)DbcH(τ)=∑iCa,iCb,σ(i)∑jCb,jCc,τ(j)=∑i,jCa,iCb,jCb,σ(i)Cc,τ(j)=∑i,jCa,iCb,jCb,σ(i)Cc,τ(j)=∑i,jCa,i(δj,σ(i)−1D)Cc,τ(j)=∑i,jCa,σ−1(j)Cc,τ(j)−1D∑i,jCa,iCc,τ(j)=∑jCa,jCc,τ(σ(j))=∑jCa,jCc,στ(j)=DacH(στ).We used(B.4)∑aCa,iCa,j=(δij−1D)∑aCa,i=0.Using the definition (B.1), we prove(B.5)∑bDbaH(σ)Cb,i=Ca,σ(i). Indeed(B.6)∑bDbaH(σ)Cb,i=∑jCb,jCa,σ(j)Cb,i=∑jCa,σ(j)(δi,j−1D)=Ca,σ(i). It is useful to define Cσ(a),i=∑bDHba(σ)Cbi so the above can be expressed as an equivariance property(B.7)Cσ(a),i=Ca,σ(i), which is an equivariance condition for the map VH→Vnat given by the coefficients Ca,i. This map intertwines the Sn action on the VH and Vnat. Now define Ca,b,c(B.8)Ca,b,c=∑iCa,iCb,iCc,i. We show that this is an invariant tensor.(B.9)Cσ(a),σ(b),σ(c)=Ca,b,c. Indeed(B.10)Cσ(a),σ(b),σ(c)=∑i=1nCσ(a),iCσ(b),iCσ(c),i=∑iCa,σ(i)Cb,σ(i)Cc,σ(i)=∑iCa,iCb,iCc,i=Ca,b,c. We used the equivariance of the C's, then the relabelled the sum i→σ(i).Using vectors {ea} spanning VH, we write a basis for the tensor product VH⊗VH:(B.11)ea⊗eb. There is a subspace of VH⊗VH, which transforms as the irrep VH. This is constructed using the invariant 3-index tensor Ca,b,c. The linear combinations(B.12)Ea=∑a,b,cCa,b,ceb⊗ec span the subspace VH in the direct sum decomposition of VH⊗VH (Equation (2.27)) under the diagonal action of SD. To see this, we can write the diagonal action of σ∈SD(B.13)σEa=∑a,b,cCa,b,c(σeb)⊗(σec)=∑a,b,c∑b′,c′Ca,b,cDb′bH(σ)Dc′cH(σ)(eb′⊗ec′)=∑a,b,c∑d,b′,c′Cd,b,cDHad(σ−1σ)Db′bH(σ)Dc′cH(σ)(eb′⊗ec′)=∑a,b,c∑a′,d,b′,c′Cd,b,cDaa′H(σ−1)Da′dH(σ)Db′bH(σ)Dc′cH(σ)(eb′⊗ec′)=∑a,a′,b′,c′Cσ−1(a′),σ−1(b′),σ−1(c′)Da′,aH(σ)(eb′⊗ec′)=∑a′Da′,aH(σ)∑a,b,cCa′,b′,c′(eb′⊗ec′)=∑a′Da′aH(σ)Ea′, showing that the transformation is indeed by the matrix DH.These vectors Ea are orthogonal. It is useful to calculate the inner product(B.14)(Ea1,Ea2)=∑b,cCa1,b,cCa2,b,c=∑i,j∑b,cCa1,iCb,iCc,iCa2,jCb,jCc,j=∑i,jCa1,iCa2,j(δi,j−1D)(δi,j−1D)=(1−2D)∑iCa1,iCa2,i=(D−2)Dδa1,a2, which will be useful in the next section.Appendix CClebsch-Gordan coefficients and normalizationsThe normalized Clebsch-Gordan coefficients for an orthonormal basis of a subspace of VH⊗VH transforming as an irrep V obey the condition(C.1)∑a,bCa,b,cH,H→VCa,b,c′H,H→V=δcc′. This means that(C.2)∑a,b,cCa,b,cH,H→VCa,b,cH,H→V=DimV.If instead we consider the invariant state in H⊗H⊗V, normalized to one, then(C.3)∑a,b,c∑a′,b′,c′(Ca,b,cH,H,Vea⊗eb⊗ec,Ca′,b′,c′H,H,Vea′⊗eb′⊗ec′)=∑a,b,c(Ca,b,cH,H,V)2=1.The equivariance property of the map CH,H→V is(C.4)DH⊗H(σ⊗σ)∑a,bCa,b,cH,H→Vea⊗eb=∑b′,c′Da′aH(σ)Db′bH(σ)Ca,b,cH,H→Vea′⊗eb′=∑c′Dc′cH(σ)∑a′,b′Ca′,b′,c′H,H→Vea′⊗eb′, which means(C.5)∑b′,c′Da′aH(σ)Db′bH(σ)Ca,b,cH,H→V=Dc′cH(σ)Ca,b,c′H,H→V. Multiplying on both sides by DceH(σ−1) and summing over c, we have, after using DabH(σ−1)=DbaH(σ) and relabelling indices(C.6)∑b′,c′Da′aH(σ)Db′bH(σ)Dc′cH(σ)Ca,b,cH,H→V=Ca,b,cH,H→V. This means that we can identify(C.7)Ca,b,cH,H→V=DimVCa,b,cH,H,V.We also know that(C.8)Ca,b,c=∑iCa,iCb,iCc,i, has the invariance property of Ca,b,cH,H,H. Since there is a unique invariant state in VH⊗VH⊗VH, Cabc must be proportional to Ca,b,cH,H,H. We calculate(C.9)∑a,b,cCa,b,cCa,b,c=∑i,j∑a,b,cCa,iCb,iCc,iCa,jCb,jCc,j=∑i,j(δij−1D)(δij−1D)(δij−1D)=1×D−3D×D+3D2×D−1D3×D2=D−3+2D−1=D−1(D−1)(D−2). We can therefore identify(C.10)Ca,b,cH,H,H=D(D−1)(D−2)CabcCa,b,cH,H→H=D(D−2)Ca,b,c. The second equation is also consistent, as expected, with (B.14). The projector for VH in VH⊗VH is given in terms of the normalized Clebsch-Gordan coefficients Ca,b,cH,H→H as(C.11)Pab;cdH,H→H=∑dCa,b,eH,H→HCc,d,eH,H→H.References[1]Z.HarrisMathematical Structures of Language1968WileyZ. Harris, “Mathematical Structures of Language,” Wiley, 1968[2]J.R.FirthA synopsis of linguistic theory 1930-1955Studies in Linguistic Analysis1957J.R.Firth, “A Synopsis of Linguistic Theory 1930-1955,” Studies in Linguistic Analysis, 1957.[3]B.CoeckeM.SadrzadehS.ClarkMathematical foundations for a compositional distributional model of meaningLambek FestschriftLinguist. Anal.362010345384B. Coecke, M. Sadrzadeh, and S. Clark “Mathematical Foundations for a Compositional Distributional Model of Meaning”. Lambek Festschrift. Linguistic. Analysis,36,345-384, 2010[4]E.GrefenstetteM.SadrzadehConcrete models and empirical evaluations for a categorical compositional distributional model of meaningComput. Linguist.41201571118E. Grefenstette and M. Sadrzadeh, “Concrete models and empirical evaluations for a categorical compositional distributional model of meaning.” Computational Linguistics, 41:71-118.[5]J.MaillardS.ClarkE.GrefenstetteA type-driven tensor-based semantics for CCGProceedings of the Type Theory and Natural Language Semantics Workshop2014EACLJ. Maillard, S. Clark, and E. Grefenstette, “A type-driven tensor-based semantics for CCG,” In Proceedings of the Type Theory and Natural Language Semantics Workshop, EACL.[6]M.BaroniR.BernardiR.ZamparelliFrege in space: a program of compositional distributional semanticsLinguist. Issues Lang. Technol.92014M. Baroni, R. Bernardi, and R. Zamparelli, “Frege in space: A program of compositional distributional semantics. Linguistic Issues in Language Technology, 9.[7]D.KartsaklisM.SadrzadehS.PulmanA unified sentence space for categorical distributional-compositional semantics: Theory and experimentsProceedings of 24th International Conference on Computational Linguistics (COLING): PostersMumbai, India2012549558D. Kartsaklis, M. Sadrzadeh, and S. Pulman. “A unified sentence space for categorical distributional-compositional semantics: Theory and experiments.” In Proceedings of 24th International Conference on Computational Linguistics (COLING): Posters, pages 549-558, Mumbai, India.[8]D.KartsaklisS.RamgoolamM.SadrzadehLinguistic matrix theoryarXiv:1703.10252 [cs.CL]2019Ann. Inst. Henri Poincaré D201910.4171/AIHPD/75D. Kartsaklis, S. Ramgoolam and M. Sadrzadeh, “Linguistic Matrix Theory,” arXiv:1703.10252 [cs.CL], to appear in Annales de l'Institut Henri Poincare D.[9]D.KartsaklisS.RamgoolamM.SadrzadehLinguistic matrix theory(short version) Quantum Physics and Logic Conference 2017, pdf available athttp://qpl.science.ru.nl/accepted.htmlVideo athttps://www.youtube.com/watch?v=RAractFNESUD. Kartsaklis, S. Ramgoolam and M. Sadrzadeh, “Linguistic Matrix Theory,” (short version) Quantum Physics and Logic Conference 2017, pdf available at http://qpl.science.ru.nl/accepted.html; Video at https://www.youtube.com/watch?v=RAractFNESU[10]https://oeis.org/A052171https://oeis.org/A052171[11]M.HamermeshGroup Theory and Its Application to Physical Problems1962DoverM. Hamermesh, “Group theory and its application to physical problems,” Dover 1962.[12]W.FultonJ.HarrisRepresentation Theory: A First Course2004SpringerW. Fulton and J. Harris, “Representation Theory: a first course” Springer 2004.[13]A.ZeeGroup Theory in a Nutshell for Physicists2016Princeton University PressA. Zee, “Group Theory in a Nutshell for Physicists,” Princeton University Press, 2016.[14]M.A.NaimarkA.I.SternTheory of Group RepresentationsA Series of Comprehensive Studies in Mathematicsvol. 2461982M. A. Naimark, A. I. Stern, “Theory of Group Representations,” A series of Comprehensive Studies in Mathematics, 246.[15]R.de Mello KochS.RamgoolamFree field primaries in general dimensions: counting and construction with rings and modulesJ. High Energy Phys.1808201808810.1007/JHEP08(2018)088arXiv:1806.01085 [hep-th]R. de Mello Koch and S. Ramgoolam, “Free field primaries in general dimensions: Counting and construction with rings and modules,” JHEP 1808 (2018) 088 doi:10.1007/JHEP08(2018)088 [arXiv:1806.01085 [hep-th]].[16]Online notes by David Zhanghttp://david-k-zhang.com/notes/gaussian-integrals.htmlOnline notes by David Zhang: http://david-k-zhang.com/notes/gaussian-integrals.html[17]https://en.wikipedia.org/wiki/Common_integrals_in_quantum_field_theoryhttps://en.wikipedia.org/wiki/Common_integrals_in_quantum_field_theory[18]Y.H.HeV.JejjalaB.D.Nelsonhep-tharXiv:1807.00735 [cs.CL]Y. H. He, V. Jejjala and B. D. Nelson, “hep-th,” arXiv:1807.00735 [cs.CL].[19]GeorgiaBenkartTomHalversonPartition algebras and the invariant theory of the symmetric grouparXiv:1709.07751 [math.RT]Georgia Benkart, Tom Halverson, “Partition Algebras and the Invariant Theory of the Symmetric Group,” arXiv:1709.07751 [math.RT][20]F.GabrielCombinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalizationhttps://arxiv.org/abs/1503.02792F. Gabriel, “Combinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalization,” https://arxiv.org/abs/1503.02792[21]F.GabrielCombinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processeshttps://arxiv.org/abs/1507.02465F. Gabriel, “Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes,” https://arxiv.org/abs/1507.02465[22]BensonAuGuillaumeCébronAntoineDahlqvistFranckGabrielCamilleMaleLarge permutation invariant random matrices are asymptotically free over the diagonalhttps://arxiv.org/abs/1805.07045Benson Au, Guillaume Cébron, Antoine Dahlqvist, Franck Gabriel, Camille Male “Large permutation invariant random matrices are asymptotically free over the diagonal,” https://arxiv.org/abs/1805.07045[23]CamilleMaleTraffic distributions and independence: permutation invariant random matrices and the three notions of independencehttps://arxiv.org/abs/1111.4662Camille Male, “Traffic distributions and independence: permutation invariant random matrices and the three notions of independence,” https://arxiv.org/abs/1111.4662[24]M.PeskinD.V.SchroderAn Introduction to Quantum Field Theory1995Taylor and Francis GroupM. Peskin, D. V. Schroder, “An introduction to quantum field theory,” Taylor and Francis Group, 1995.[25]A.ZeeQuantum Field Theory in a Nutshell2010Princeton University PressA. Zee, “Quantum Field Theory in a nutshell,” Princeton University Press, 2010.[26]https://en.wikipedia.org/wiki/Isserlis%27_theoremhttps://en.wikipedia.org/wiki/Isserlis%27_theorem[27]G.'t HooftA planar diagram theory for strong interactionsNucl. Phys. B72197446110.1016/0550-3213(74)90154-0G. 't Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72 (1974) 461. doi:10.1016/0550-3213(74)90154-0[28]J.M.MaldacenaThe large N limit of superconformal field theories and supergravityInt. J. Theor. Phys.381999111310.1023/A:1026654312961Adv. Theor. Math. Phys.2199823110.4310/ATMP.1998.v2.n2.a1arXiv:hep-th/9711200J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] doi:10.1023/A:1026654312961, 10.4310/ATMP.1998.v2.n2.a1 [hep-th/9711200].