Quantum spins and random loops on the complete graph

We present a systematic analysis of quantum Heisenberg-, XY- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson--Dirichlet distributions.

operator". For these models, we investigate the structure of the space, Ψ β , of extremal Gibbs states at inverse temperature β = (kT ) −1 , for different values of β. Following a suggestion of Thomas Spencer, we analyze the generating function, Φ β (h), of correlations of the averaged spin density in the symmetric Gibbs state at inverse temperature β, which depends on a symmetry-breaking external magnetic field, h. The function Φ β (h) can be viewed as a Laplace transform of the measure dµ on Ψ β whose barycenter is the symmetric Gibbs state at inverse temperature β. Its usefulness lies in the fact that it sheds light on the structure of the space of extremal Gibbs states. We calculate Φ β (h) explicitly for a class of (mean-field) spin models defined on the complete graph, for all values of β > 0. It is expected that the dependence of Φ β (h) on the external magnetic field h is universal, in the sense that it is equal to the one calculated for the corresponding models defined on the lattice Z d , provided the dimension d satisfies d ≥ 3. Moreover, the structure of Ψ β is expected to be independent of d, for d ≥ 3, and identical to the one in the models on the complete graph. Rigorous proofs, however, still elude us. The quantum spin systems studied in this paper happen to admit random loop representations, and the functions Φ β (h) correspond to characteristic functions of the lengths of random loops. It turns out that these characteristic functions are equal to those of the Poisson-Dirichlet distribution of random partitions. This is a strong indication that the joint distribution of the lengths of the random loops is indeed the Poisson-Dirichlet distribution.
Next, we briefly review the general theory of extremal-states decompositions. (For more complete information we refer the reader to the 1970 Les Houches lectures of the late O. E. Lanford III [18], and the books of R. B. Israel [14] and B. Simon [26].) The set, G β , of infinitevolume Gibbs states at inverse temperature β forms a Choquet simplex, i.e., a compact convex subset of a normed space with the property that every point can be expressed uniquely as a convex combination of extreme points, (i.e., as the barycenter of a probability measure supported on extreme points). As above, let Ψ β ⊂ G β denote the space of extremal Gibbs states at inverse temperature β. Henceforth we denote an extremal Gibbs state by · ψ , with ψ ∈ Ψ β . Since G β is a Choquet simplex, an arbitrary state · ∈ G β determines a unique probability measure dµ on Ψ β such that · = Ψ β · ψ dµ(ψ). (1.1) At small values of β, i.e., high temperatures, the set G β of Gibbs states at inverse temperature β contains a single element, and the above decomposition is trivial. The situation tends to be more interesting at low temperatures: the set G β may then contain many states, in which case one would like to characterise the set Ψ β of extreme points of G β .
In the models studied in this paper, the Hamiltonian is invariant under a continuous group, G, of symmetries, and the set G β of Gibbs states at inverse temperature β carries an action of the group G. At low temperatures, this action tends to be non-trivial; i.e., there are plenty of Gibbs states that are not invariant under the action of G on G β . This phenomenon is referred to as "spontaneous symmetry breaking". For the models studied in this paper, the space Ψ β of extremal Gibbs states is expected to consist of a single orbit of an extremal state · ψ0 , ψ 0 ∈ Ψ β , under the action of G (this is clearly a special case of the general situation). Then Ψ β G/H, where H is the largest subgroup of G leaving · ψ0 invariant, and the symmetric (i.e., G-invariant) state in G β can be obtained by averaging over the orbit of the state · ψ0 under the action of the group G using the (uniform) Haar measure on G.
As announced above, we will follow a suggestion of T. Spencer and attempt to characterise the set Ψ β by considering a Laplace transform Φ β (h) of the measure on Ψ β whose barycenter is the symmetric state. We describe the general ideas of our analysis for models of quantum spin systems defined on a lattice Z d , d ≥ 3; afterwards we will rigorously study similar models defined on the complete graph. At each site i ∈ Z d , there are N operators S i = (S (1) , . . . , S (N ) ) describing a "quantum spin" located at the site i. We assume that the symmetry group G is represented on the algebra of spin observables generated by the operators { S i | i ∈ Z d } by * -automorphisms, α g , g ∈ G, with the property that there exist N × N -matrices R(g), g ∈ G, acting transitively on the unit sphere S N −1 ⊂ R N such that α g ( S · n) = S · R(g) n, ∀ n ∈ R N . (1.2) We assume that the states · ψ , ψ ∈ Ψ β , are invariant under lattice translations. Denoting by · Λ,β the symmetric Gibbs state in a finite domain Λ ⊂ Z d , and by Λ ⇑ Z d the standard infinite-volume limit (in the sense of van Hove), we consider the generating function 0 is the spin operator S (1) acting at the site 0. The first identity is expected to hold true in great generality; but it appears to be difficult to prove it in concrete models. The second identity holds under very general assumptions, but the exact structure of the space Ψ β and the properties of the measure dµ are only known for a restricted class of models, such as the Ising-and the classical xy-model. The third identity usually follows from cluster properties of connected correlations in extremal states.
Assuming that all equalities in (1.3) hold true, we define the ("spin-density") Laplace transform of the measure dµ corresponding to the symmetric state by The action of G on the space G β of Gibbs states is given by for an arbitrary spin observable A. As mentioned above, we will consider models for which it is expected that Ψ β is the orbit of a single extremal state, · ψ0 ; i.e., given ψ ∈ Ψ β , there exists an element g(ψ) ∈ G such that ψ0 , (1.5) where g(ψ) is unique modulo the stabilizer subgroup H of · ψ0 . Then we have that (1.6) Defining the magnetisation as m d (β) = S 0 ψ0 , we find that the spin-density Laplace transform (1.4) is given by where e 1 is the unit vector in the 1-direction in R N ; (actually, e 1 can be replaced by an arbitrary unit vector in R N ).
In this paper we study a variety of quantum spin systems for which we will calculate the function Φ β (h) in two different ways: (1) For an explicit class of models defined on the complete graph, we are able to calculate the function Φ β (h) explicitly and rigorously. (2) On the basis of some assumptions on the structure of the set Ψ β of extremal Gibbs states and on the matrices R(g), g ∈ G, that we will not justify rigorously, we are able to determine Φ β (h) using (1.3). We then observe that the two calculations yield identical results, representing support for the assumptions underlying calculation (2).
Organization of the paper. In Section 2 we provide precise statements of our results and verify that they are consistent with the heuristics captured in Eq. (1.3). In Section 3 we describe (known) representations of the spin systems considered in this paper in terms of random loops; we then discuss probabilistic interpretations of our results and relate them to the Poisson-Dirichlet distribution. In Sections 4 through 7, we present proofs of our results. Some auxiliary calculations and arguments are collected in four appendices.

Setting and results
In this section we describe the precise setting underlying the analysis presented in this paper. Rigorous calculations will be limited to quantum models on the complete graph.
Let n ∈ N be the number of sites, and let S ∈ 1 2 N be the spin quantum number. The state space of a model of quantum spins of spin S located at the sites {1, . . . , n} is the Hilbert space H n = (C 2S+1 ) ⊗n . The usual spin operators acting on H n are denoted by with further commutation relations obtained by cyclic permutations of 1,2,3; furthermore, The Hamiltonian, H Heis n,∆ , of the quantum Heisenberg model is given by 3) The value ∆ = 0 corresponds to the xy-model, and ∆ = 1 corresponds to the usual Heisenberg ferromagnet. By · Heis n,β,∆ we denote the corresponding Gibbs state · Heis n,β,∆ = The Hamiltonian of the quantum interchange model is chosen to be 5) where the operators T i,j are the transposition operators defined by T i,j |ϕ 1 · · · ⊗ |ϕ i · · · ⊗ |ϕ j · · · ⊗ |ϕ n = |ϕ 1 · · · ⊗ |ϕ j · · · ⊗ |ϕ i · · · ⊗ |ϕ n , (2.6) where the vectors |ϕ i belong to the space C 2S+1 , for all i = 1, 2, . . . , n. The transposition operators are invariant under unitary transformations of C 2S+1 and can be expressed using spin operators; see [21] or [8, Appendix A] for more details. Recall that the eigenvalues of ( S i + S j ) 2 are given by λ(λ + 1), with λ = 0, 1, . . . , 2S; hence the eigenvalues of 2 S i · S j are given by λ(λ + 1) − 2S(S + 1). Denoting by P λ the corresponding spectral projections we find that It is apparent that T i,j is a linear combination of ( S i · S j ) k , with k = 0, 1, . . . , 2S. One checks that 2 the quantum interchange model is equivalent to the Heisenberg ferromagnet, but this is not the case for other values of the spin quantum number S. (The expressions for T i,j , with S ≥ 3 2 , look unappealing.) The Gibbs state of the quantum interchange model is given by Tr [· e −βH int n ] .
(2.9) 2.1. Heisenberg and xy-models. First we consider the Heisenberg model with ∆ = 1 and arbitrary spin S ∈ 1 2 N. In order to define the spontaneous magnetisation, we introduce a function η : R → R by setting . (2.10) (At x = 0 we define η(0) = log(2S + 1).) Its first and second derivatives are . (2.11) Note that this function is smooth at x = 0, where η (0) = 0. The second derivative is positive, and η (±∞) = ±S, so that the equation η (x) = m, (2.12) has a unique solution for all m ∈ (−S, S). We denote this solution by x (m). Lengthy calculations yield Next, we define a function g β by (2.14) One finds that g β (0) = log(2S + 1); g β (0) = 0; and g β (0) = 2β − 3 S 2 + S . (2.15) Let m (β) ∈ [0, S) be the maximiser of g β . From (2.15) we infer that m (β) > 0 if and only if β is greater than the critical inverse temperature β c given by β c = 3/2 S 2 + S . (2.16) It may be useful to note that, for S = 1 2 , the above definitions simplify considerably: (2.17) One easily checks that g β (0) = 0, g β (m) < 0 for all m ∈ (0, 1 2 ), and that g β (0) = 2β − 4 is positive if and only if β > 2. It follows that the unique maximiser m (β) is positive if and only if β > 2; see Fig. 1. For the symmetric spin-1 2 Heisenberg model (S = 1 2 and ∆ = 1), the magnetisation m (β) was first identified by Tóth [29] and Penrose [23]. (See also the recent paper [3] by Alon and Kozma.) The proof of this theorem can be found in Section 4. Concerning symmetry breaking, we expect that the extremal states are labeled by a ∈ S 2 . (The 2-sphere is the orbit of any point on Ψ β under the action of the symmetry group SO(3), and H = SO(2)). For a ∈ S 2 we introduce the following Gibbs states: , · a = lim h↓0 · a,h . (2.18) For h = 0 the states · a,h are extremal by an extension of the Lee-Yang theorem [4,28]; it is reasonable to expect that the limiting states · a are also extremal, although this has not been proved. (A non-trivial technical issue is whether the limits in (2.18) exist; but we do not worry about it in this discussion.) Defining m (β) = S (1) i e1 , we have that (2.19) where e 1 = (1, 0, 0) T is the unit vector in the 1-direction. Assuming that (1.3) is correct, we expect that (2.20) The right side of (2.20) coincides with the expression in Theorem 2.1; so (1.3) is expected to be correct for this model. Our next result concerns the Heisenberg Hamiltonians with ∆ ∈ [−1, 1). Models with these Hamiltonians behave just like the xy-model, (∆ = 0). For models on the complete graph, this remains true also for ∆ = −1. (However, on a bipartite graph (lattice), the model with ∆ = −1 is unitarily equivalent to the quantum Heisenberg antiferromagnet whose properties are different from those of the xy-model.) We let m (β) be the maximiser of the function g β in (2.14), as before. Let I 0 (x) = k≥0 1 (k!) 2 ( x 2 ) 2k be the modified Bessel function.
The proof of this theorem can be found in Section 5. This theorem confirms that the phase transition signals the onset of spontaneous magnetisation in the 1-2 plane. We now introduce , for a ⊥ e 3 , | a| = 1 .  In order to define the object that plays the rôle of the magnetisation, let φ β be the function [0, 1] 2S+1 → R given by We look for maximisers (x 1 , . . . , x 2S+1 ) of φ β under the condition i x i = 1 and x 1 ≥ x 2 ≥ · · · ≥ x 2S+1 . It was understood and proven by Björnberg, see [8,Theorem 4.2], that the answer involves the critical parameter The maximiser is unique and satisfies (see Appendix C). The analogue of the magnetisation is defined as (2. 26) In the following theorem, R denotes the function and if A is an arbitrary (2S + 1) × (2S + 1) matrix then A i := 1l ⊗ · · · ⊗ A ⊗ · · · ⊗ 1l, where A occupies the ith factor. Note that R is continuous: in the numerator, det e hixj θ i,j=1 is analytic in the variables h i and x i , and it is anti-symmetric under permutations of the arguments h i and x i , hence it vanishes whenever two or more of the h i 's or of the x i 's coincide.
We highlight the following two special cases of this result: first, we get that ; (2.28) second, if Q denotes an arbitrary rank 1 projector, with eigenvalues 1, 0, . . . , 0, we get The step from Theorem 2.3 to (2.28) and (2.29) is not immediate; details appear in Sect. 6. Next, we discuss the heuristics of spontaneous symmetry breaking. The Hamiltonian of the interchange model is invariant under an SU(2S + 1)-symmetry: Given an arbitrary As pointed out to us by Robert Seiringer, the extremal states are labeled by rank-1 projections on C 2S+1 , or, equivalently, by the complex projective space CP 2S (i.e., by the set of equivalence classes of vectors in C 2S+1 only differing by multiplication by a complex nonzero number). Given v ∈ C 2S+1 \{0}, let P v denote the orthogonal projection onto v, and let P v i := 1l⊗· · ·⊗P v ⊗· · ·⊗1l, where P v occupies the ith factor. The extremal states are expected to be given by . (2.30) As β → ∞, · v converges to the expectation defined by the product state ⊗v. These product states are ground states of H int n , which gives some justification to the claim that the states · v are extremal. We expect that We take the state · e1 as the reference state, with vector v = e 1 = (1, 0, . . . , 0). At the cost of some redundancy, the integral over v in CP 2S can be written as an integral over the space U(2S + 1) of unitary matrices on C 2S+1 with the uniform probability (Haar) measure: (2.32) Next we consider the restriction of the state · e1 onto operators that only involve the spin at site 1. This restriction can be represented by a density matrix ρ on C 2S+1 such that In all bases where e 1 = (1, 0, . . . , 0), the matrix ρ is diagonal with entries (x 1 , . . . , x 2S+1 ) on the diagonal, where x i = Tr (P ei ρ) = P ei 1 e1 . (2.34) It is clear that x 2 = · · · = x 2S+1 , and one should expect that x 1 is larger than or or equal to x * 2 . Heuristic arguments suggest that By the Harish-Chandra-Itzykson-Zuber formula [15], the right-hand-side of (2.35) is equal to R(h 1 , . . . , h 2S+1 ; x 1 , . . . , x 2S+1 ) which agrees with the right-hand-side in Theorem 2.3.

2.3.
Critical exponents for the Heisenberg model. Relatively minor extensions of our calculations for the Heisenberg model (∆ = 1) enable us to determine some critical exponents for that model on the complete graph. To state our results, we introduce the pressure (more accurately, this is (−β) times the free energy; "pressure" is used by analogy to the Ising model, where it is justified by the lattice-gas interpretation). Next, we consider the magnetization and susceptibility and the transverse susceptibility The following theorem is proven in Section 7. Recall the function g β (m), 0 ≤ m ≤ S, given in (2.14) (which reduces to (2.17) for S = 1 2 ). We write f ∼ g if f /g converges to a positive constant.
Theorem 2.4. For the spin-S ≥ 1 2 Heisenberg models the following formulae hold true.
(i) Pressure: and We note that the critical exponents (2.40) are exactly the same as for the classical spin- → cosh(hm ). In proving (2.41) we will use general inequalities relating the transverse susceptibility to the magnetization, which follow from Ward-identities and the Falk-Bruch inequality. For details, see Section 7.

Random loop representations
The Gibbs states of quantum spin systems can be described with the help of Feynman-Kac expansions. In some cases these expansions can be represented as probability measures on sets of loop configurations. Such cases include Tóth's random interchange representation for the spin-1 2 Heisenberg ferromagnet. (An early version of this representation is due to Powers [24]; it was independently proposed by Tóth in [30], with a precise formulation and interesting applications.) Another useful representation is Aizenman and Nachtergaele's loop model for the spin-1 2 Heisenberg antiferromagnet, and models of arbitrary spins where interactions are given by projectors onto spin singlets [1]. Nachtergaele extended these representations to Heisenberg models of arbitrary spin [21]. A synthesis of the Tóth-and the Aizenman-Nachtergaele loop models, which allows one to describe the spin-1 2 xy-model and a spin-1 nematic model, was proposed in [32].
These models are interesting from the point of view of probability theory and they are relevant here because the joint distribution of loop lengths turns out to be related to the extremal state decomposition of the corresponding quantum systems. Indeed, some characteristic functions for the loop lengths are equal to the Laplace transforms of the measure on the set of extremal states.
The loop models considered in this paper can be defined on any graph Γ, and involve onedimensional loops immersed in the space Γ × [0, β]. Quantum-mechanical correlations can be expressed in terms of probabilities for loop connectivity. The lengths of the loops, rescaled by an appropriate fractional power of the spatial volume, are expected to display a universal behavior: there are macroscopic and microscopic loops, and the limiting joint distribution of the lengths of macroscopic loops is expected to be the Poisson-Dirichlet (PD) distribution that originally appeared in the work of Kingman [16]. This distribution is illustrated in The Poisson-Dirichlet distribution, denoted by PD(θ), with θ > 0 arbitrary, can be defined via the following 'stick-breaking' construction: The vector X obtained by ordering the elements of Y by size has the PD(θ)-distribution. Note that i≥1 X i = 1 with probability 1, hence the X i may be regarded as giving a partition of the interval [0, 1]. To obtain a partition of an interval [0, z ] as in Fig. 2 one simply rescales X by z . For future reference we note here the following formula, which will turn out to be relevant for the spin-systems considered in this paper. In [34,Eq. (4.18)] it is shown that The Poisson-Dirichlet distribution first appeared in the study of the random interchange model (transposition-shuffle) on the complete graph. David Aldous formulated a conjecture concerning the convergence of the rescaled loop sizes to PD(1), and he explained the heuristics; Schramm then provided a proof [25] of Aldous' conjecture. Models on the complete graph are easier to analyse than the corresponding models on a lattice Z d , d ≥ 3; but the heuristics for the latter models is remarkably similar to the one for the former models; see [12,34]. The ideas sketched here are confirmed by the results of numerical simulations of various loop soups, including lattice permutations [13], loop O(N)-models [22], and the random interchange model [5].
3.1. Spin-1 2 models. We begin by describing the loop representations of the Heisenberg models with spin S = 1 2 . These representations are quite well known and contain many of the essential features, but without some of the complexities that appear for larger spin.
We pick a real number u ∈ [0, 1]. Let Γ = K n be the complete graph, with vertices V n = {1, . . . , n} and edges E n = {i, j} : 1 ≤ i < j ≤ n . With each edge we associate an independent Poisson point process on the time interval [0, β/n] with two kinds of outcomes: 'crosses' occur with intensity u and 'double bars' occur with intensity 1 − u. We let ρ n,β,u denote the law of the Poisson point processes. Given a realization ω, the loop containing the point (v, t) ∈ K n × [0, β/n] is obtained by moving vertically until meeting a cross or a double bar, then crossing the edge to the other vertex, and continuing in the same vertical direction, for a cross, while continuing in the opposite direction, for a double bar; see where the normalisation Z(n, β, 2, u) = 2 |L(ω)| ρ n,β,u (dω) is the partition function. By E n,β,2,u we denote an expectation with respect to this probability measure. We define the length of a loop as the number of points (i, 0) that it contains; i.e., the length of a loop is the number of sites at level 0 ∈ [0, β/n] visited by the loop. (According to this definition, there are loops of length 0.) Given a realisation ω, let 1 (ω), 2 (ω), . . . be the lengths of the loops in decreasing order. We have that i≥1 i (ω) = n, for an arbitrary ω.
One manifestation of the connection between the loop-model and the spin system is the following identity, valid for ∆ = 2u − 1: This is a special case of (3.19) below.

3.2.
Heisenberg models with arbitrary spins. An extension of the loop representation for the Heisenberg ferromagnet (and antiferromagnet, and further interactions) with arbitrary spin was proposed by Bruno Nachtergaele [21]. As in [32] it can be generalised to include asymmetric Heisenberg models. We first describe this representation and state our results about the lengths of the loops. Afterwards, we will outline the derivation of this representation from models of spins. We introduce a model where every site is replaced by 2S "pseudo-sites". Let K n be the graph whose vertices are the pseudo-sites (i, α) : i ∈ {1, . . . , n}, α ∈ {1, . . . , 2S} and whose edges are given by We require the following ingredients: • A uniformly random permutation σ of the pseudo-sites at each vertex; namely, σ = (σ i ) n i=1 , where the σ i are independent, uniform permutations of 2S elements. Let ρ n,β,u denote the measure for the Poisson point process. The measure on the set of permutations is just the counting measure. Loops are defined as before, except that the permutations rewire the threads between times β 2n and − β 2n . An illustration is given in Fig.  4.
The probability measure relevant for the following considerations is the following measure: P n,β,2,u (σ, dω) = 1 Z(n, β, 2, u) 2 |L(σ,ω)| ρ n,β,2,u (dω). Expectation with respect to P n,β,2,u (σ, dω) is denoted byẼ n,β,2,u . We define the length of a loop as the number of sites at time 0 visited by it. For any realisation (σ, ω), we have that i≥1 i (σ, ω) = 2Sn. Figure 4. Loop representation for Heisenberg models with spins S = 3 2 . The original graph is modified so each site is now hosting 2S = 3 pseudosites. There are random permutations of pseudo-sites between times β 2n and − β 2n . As before, there is an overall factor 2 #loops . In the realisation above, one loop is highlighted (it has length 3) and there are three other loops (of length 0, 4, and 5).
As we will explain below, this loop model provides a probabilistic representation of the Heisenberg model with ∆ = 2u − 1. The two parts of the following theorem are equivalent to Theorems 2.1 and 2.2, respectively.
We note that the limiting quantities agree with the corresponding expectations with respect to the Poisson-Dirichlet distributions; more precisely PD(2), for u = 1, and PD(1), for u < 1. Indeed, setting θ = 2 in (3.2), we find that

7)
while setting θ = 1 yields Next, we explain how to derive this loop model from quantum spin systems. This will show that Theorem 3.1 is equivalent to Theorem 2.1.

It immediately follows from this proposition that
(3. 19) In particular, Theorem 3.1 follows from Theorems 2.1 and 2.2, which are proven in Sects. 4 and 5, respectively.
With this result in hand, the proof of Theorem 2.1 is straightforward: Proof of Theorem 2.1. We will write · for · Heis n,β,∆=1 .We assume that Sn is an integer; (the case of half-integer values being almost identical). Using Proposition A.1, we get  as claimed.
Remark 4.2. Letting S → ∞ in Theorem 2.1, with the appropriate rescaling h → h/S and β → β/S 2 , and using the results of Lieb [19] we recover the corresponding generating function for the classical Heisenberg model. The limit is sinh(hµ )/hµ where µ ∈ [0, 1] is the maximizer of
Assuming that |J/n − m | < ε and that |M/n| < ε, the last product in (5.8) is seen to be bounded by We first consider a range of temperatures with the property that m (β) = 0. It then follows from a rather crude estimate that The sum on the right side of this inequality is uniformly convergent, provided ε is small enough and n is large enough. It can be made arbitrarily small by choosing ε small enough and n large enough. It follows that, under the assumption that m = 0, A(J, n) is of the form A(J, n) = 1 + ε 2 (J, n), with ε 2 → 0, as n → ∞, uniformly in J. By Lemma B.1, this completes our proof for the case that m = 0. Next, we consider the range of temperatures with m (β) > 0. We pick a sufficiently small ε < m . The number of sequences (δ i ) k i=1 satisfying the constraints in (5.8) is bounded by  (5.12) One can check that the sum on the right side of this inequality converges uniformly in n, for n large enough. It can be made as small as we wish by choosing ε small enough and n large enough.
To prove a lower bound, we take K so large that k>K Continuing to assume that |J/n − m | < ε and |M/n| < ε, we find that the number of sequences (δ i ) k i=1 satisfying the constraints in (5.8) equals k k/2 , provided that k ≤ K < (m − 2ε)n. The last product in (5.8 Taking n large enough and ε small enough, the sum on the right side of this inequality can be made as small as we wish. This proves that A(J, n) = I 1 (hm )/( 1 2 hm ) + ε 2 (J, n), for some ε 2 → 0, uniformly in J. This completes the proof of our claim.

Interchange Model -Proof of Theorem 2.3
When studying the interchange model we prefer to use the probabilistic representation in our proof. Thus we prove the statements in Theorem 3.3, which is equivalent to Theorem 2.3. Our proof relies on the fact that the loop-representation involves random walks on the symmetric group S n . For this reason, there are (group-) representation-theoretic tools available to analyse our models. Specifically we will make use of tools developed by Alon, Berestycki and Kozma [2,6]. A similar approach has been followed in [8] in a calculation of the free energy and of the critical point of the model. In this section, we will also use the connection between representations of S n and symmetric polynomials.
Next, we summarise some relevant facts about symmetric polynomials and representations of S n ; see [20,Ch. I] or [27,Ch. 7], for more information. By a partition we mean a vector λ = (λ 1 , λ 2 , . . . , λ k ) consisting of integer-entries satisfying λ 1 ≥ λ 2 ≥ · · · λ k ≥ 1. If j λ j = n then we say that λ is a partition of n and we write λ n. We call (λ) = k the length of λ, and if j > (λ) we set λ j = 0. We consider two types of symmetric polynomials in the variables x = (x 1 , . . . , x r ). We begin by defining the power-sums Next, we define the Schur-polynomials Note that s λ (x) is indeed a polynomial: the determinant in the numerator is a polynomial in the variables x i which is anti-symmetric under permutations of the variables, hence divisible (in Z[x 1 , . . . , x r ]) by 1≤i<j≤r (x i − x j ). In particular, s λ (·) is continuous when viewed as a function C r → C.
By continuity of the Schur-polynomials we have that (6.8) where we use the notation 0 = (0, . . . , 0). Recall the definition of the function R from Theorems 2.3 and 3.3.
Lemma 6.1. Consider a sequence of partitions λ n such that λ/n → (x 1 , . . . , x θ ). Then, for any fixed h, we have thatf Proof. Let ε j = θ−j n + (λ j /n − x j ), so ε j → 0 as n → ∞ for all j. The left-hand-side of (6.9) equals . (6.10) Indeed, the identity holds whenever all the h i are different. Hence by continuity of the left side and of the function R it holds in general if we adopt the rule that any factor in the last product on the right side is interpreted as = 1 if h i = h j . Since R is continuous and the product converges to 1, as n → ∞, the result follows.
Let us now show how to deduce form these results the special cases (3.25) and (3.26), (which are equivalent to (2.28) and (2.29)). For (3.25) we set h i = h(−S + i − 1). From the Vandermonde determinant we get that 15) where we have used (θ − 1) j x j = i<j (x i + x j ). Hence the right side of (3.22), with . (6.16) Here, all factors with 2 ≤ i < j ≤ θ equal 1. We therefore get 17) as claimed.
Next we observe that (3.26) follows by applying Theorem 2.3, with h 1 = h and h 2 = h 3 = . . . = h θ = 0. The proof involves careful manipulation of some determinants; here we only outline the main steps.
To prove (2.41) we will use the following result.
For the other part we will use the Falk-Bruch inequality. First, there exists a positive measure µ on R such that Define the probability measure ν on R by dν(t) := 1 a t(e t − 1)dµ(t), (7.25) and consider the concave function φ : [0, ∞) → [0, ∞) given by By Jensen's inequality we have as claimed.
Proof of (2.41). We use Theorem 7.1 with |Γ| = n, u = 0 and J i,j = 1 n for i = j (and J i,i = 0). Note that M Γ (β, h) → m(β, h) as n → ∞ for h > 0, also note that we should replace βh in (7.14) by h to account for the slightly different conventions in (2.36) and (7.12).
Proof. For the first part, let α > 0 be such that x − x ≥ ε implies G(x ) ≥ G(x) + 2α, and let k satisfy k /n → x . Then for n large enough For the second part, let δ > 0 be arbitrary and let ε > 0 be such that x − x < ε implies |F (x) − F (x )| < δ. Applying the first part with A(k, n) = F (k/n) + ε 2 (k, n) − F (x ) we get for n large enough. This proves the claim.
Appendix C. Uniqueness of the maximizer of φ β Recall that, for x 1 ≥ x 2 ≥ . . . ≥ x θ ≥ 0 satisfying i x i = 1, we defined x i log x i .
(C. 1) In [8] it was proved that (for θ ≥ 3, that is S ≥ 1) φ β (·) is maximised at x 1 = x 2 = · · · = x θ = 1 θ when β < β c , and at some point satisfying x 1 > x 2 when β ≥ β c . Here we provide the following additional information about the maximiser. Proof. As noted in [8,Thm 4.2], the method of Lagrange multipliers tells us that a maximizer x of φ β (·) must be of the form x r+1 = . . . = x θ = 1−rt θ−r , Thus, when r is an integer, φ β (r, t) agrees with φ β (x) evaluated at x of the form (C.2). We aim to show: first that φ β (r, t) has no maximum in the interior of D, and second that, on the boundary ∂D, it is largest along the line r = 1. We find that ∂φ β ∂t = r β θt−1 θ−r − log t(θ−r) To look for points where both partial derivatives vanish, we put in the parameterization (C.5) and set the result to = 0. After simplifying, this reduces to the condition: which has no solution ξ > 0. It follows that any maxima of φ β (r, t) must lie on the boundary ∂D. The boundary consists of the following 3 parts: • A: the line t = 1 θ , • B: the curve t = 1 r , and • C: the line r = 1.
(C. 8) It is easy to see that f (r) is either monotone, or has only one extreme point (at r = β 2 ) which is a minimum. Thus f (r) is maximal at one of the endpoints. This proves that φ β (r, t) is maximized along C, as claimed.
For uniqueness of the maximizer note that (C.4), with r = 1, has at most two solutions ξ > 0, at most one of which can be at a maximum.