Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg-Landau Polynomials

Transcription

1 Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points via Ginzburg-Landau Polynomials Richard S. Ellis Jonathan Machta 2 machta@physics.umass.edu Peter Tak-Hun Otto 3 potto@willamette.edu Department of Mathematics and Statistics University of Massachusetts Amherst, MA Department of Physics University of Massachusetts Amherst, MA Department of Mathematics Willamette University Salem, OR 9730 July 9, 2008 Abstract The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution

2 Ellis, Machta, and Otto: Asymptotics for the Magnetization 2 that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(β n, K n ) for appropriate sequences (β n, K n ) that converge to a secondorder point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (β n, K n ) converges to one of these points (β, K), m(β n, K n ) x β β n γ 0. In this formula γ is a positive constant, and x is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg-Landau polynomial because of its close connection with the Ginzburg-Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which (β n, K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume-Capel model. American Mathematical Society 2000 Subject Classification. Primary 82B20 Key words and phrases: Ginzburg-Landau phenomenology, second-order phase transition, firstorder phase transition, tricritical point, scaling theory, Blume-Capel model Introduction In this paper we prove unexpected connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The investigation is carried out for a mean-field version of an important lattice spin model due to Blume and Capel, to which we refer as the B-C model [2, 5, 6, 7]. This mean-field model is equivalent to the B-C model on the complete graph on N vertices. It is certainly one of the simplest models that exhibit the following intricate phasetransition structure: a curve of second-order points; a curve of first-order points; and a tricritical point, which separates the two curves. A generalization of the B-C model is studied in [3]. The main result in the present paper is Theorem 3.2, a general theorem that gives the asymptotic behavior of the magnetization in the mean-field B-C model for suitable sequences. With only changes in notation, the theorem also applies to other mean-field models including the Curie-Weiss model [] and the Curie-Weiss-Potts model [5]. The mean-field B-C model is defined by a canonical ensemble that we denote by P N,β,K ; N equals the number of spins, β is the inverse temperature, and K is the interaction strength.

3 Ellis, Machta, and Otto: Asymptotics for the Magnetization 3 P N,β,K is defined in terms of the Hamiltonian H N,K (ω) = ( N ωj 2 K N ) 2 ω j, N j= j= in which ω j represents the spin at site j {, 2,..., N} and takes values in Λ = {, 0, }. The configuration space for the model is the set Λ N containing all sequences ω = (ω, ω 2,..., ω N ) with each ω j Λ. Before introducing the results in this paper, we summarize the phase-transition structure of the model. For β > 0 and K > 0 we denote by M β,k the set of equilibrium values of the magnetization. M β,k coincides with the set of global minimum points of the free-energy functional G β,k, which is defined in (2.4). It is known from heuristic arguments and is proved in [4] that there exists a critical inverse temperature β c = log 4 and that for 0 < β β c there exists a quantity K(β) and for β > β c there exists a quantity K (β) having the following properties:. Fix 0 < β β c. Then for 0 < K K(β), M β,k consists of the unique pure phase 0, and for K > K(β), M β,k consists of two nonzero values of the magnetization ±m(β, K). 2. For 0 < β β c, M β,k undergoes a continuous bifurcation at K = K(β), changing continuously from {0} for K K(β) to {±m(β, K)} for K > K(β). This continuous bifurcation corresponds to a second-order phase transition. 3. Fix β > β c. Then for 0 < K < K (β), M β,k consists of the unique pure phase 0; for K = K (β), M β,k consists of 0 and two nonzero values of the magnetization ±m(β, K (β)); and for K > K (β), M β,k consists of two nonzero values of the magnetization ±m(β, K). 4. For β > β c, M β,k undergoes a discontinuous bifurcation at K = K (β), changing discontinuously from {0} for K < K(β) to {0, ±m(β, K)} for K = K (β) to {±m(β, K)} for K > K (β). This discontinuous bifurcation corresponds to a first-order phase transition. Because of items 2 and 4, we refer to the curve {(β, K(β)), 0 < β < β c } as the secondorder curve and to the curve {(β, K (β)), β > β c } as the first-order curve. Points on the second-order curve are called second-order points, and points on the first-order curve first-order points. The point (β c, K(β c )) = (log 4, 3/2log 4) separates the second-order curve from the first-order curve and is called the tricritical point. The phase-coexistence region consists of all points in the positive β-k quadrant for which M β,k consists of more than one value. Thus

4 Ellis, Machta, and Otto: Asymptotics for the Magnetization 4 this region consists of all (β, K) above the second-order curve, above the tricritical point, on the first-order curve, and above the first-order curve; i.e., all (β, K) satisfying 0 < β β c and K > K(β) and satisfying β > β c and K K (β). The sets that describe the phase-transition structure of the model are shown in Figure. K K(β) K(β) K(β ) c.0 K (β) β c β Figure : The sets that describe the phase-transition structure of the mean-field B-C model: the second-order curve {(β, K(β)), 0 < β < β c }, the first-order curve {(β, K (β)), β > β c }, and the tricritical point (β c, K(β c )). The phase-coexistence region consists of all (β, K) above the second-order curve, above the tricritical point, on the first-order curve, and above the first-order curve. The extension of the second-order curve to β > β c is called the spinodal curve. We now turn to the main focus of this paper, which is the asymptotic behavior of the magnetization m(β n, K n ) for appropriate sequences (β n, K n ) that converge either to a second-order point or to the tricritical point from various subsets of the phase-coexistence region. In the case of second-order points we consider two such sequences in Theorems 4. and 4.2, and in the case of the tricritical point we consider four such sequences in Theorems Denoting the second-order point or the tricritical point by (β, K), in each case we prove as a consequence of the general result in Theorem 3.2 that m(β n, K n ) 0 according to the asymptotic formula m(β n, K n ) x β β n γ ; i.e. lim β β n γ m(β n, K n ) = x. (.) n In this formula γ is a positive constant, and x is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg-Landau polynomial because of its close

5 Ellis, Machta, and Otto: Asymptotics for the Magnetization 5 connection with the Ginzburg-Landau phenomenology of critical phenomena [6]. Both γ and x depend on the sequence (β n, K n ). The exponent γ and the polynomial g arise via the limit of suitably scaled free-energy functionals; specifically, for appropriate choices of u R and γ > 0 and uniformly for x in compact subsets of R lim n n u G βn,k n (x/n γ ) = g(x). (.2) Possible paths followed by the sequences studied in Theorems are shown in Figure 2. Two different paths are shown for each of the first three sequences, four different paths for the fourth sequence, and one path for each of the last two sequences. We believe that modulo uninteresting scale changes, these are all the sequences of the form β n = β + b/n α and K n equal to K(β) plus a polynomial in /n α, where (β, K(β)) is either a second-order point or the tricritical point and for which m(β n, K n ) > 0. K a 6 4b.09 K(β ) c.08 K(β) 4c.07.3 β c.4.5 K (β) 4d β Figure 2: Possible paths for the six sequences converging to a second-order point and to the tricritical point. The curves labeled, 2, 3, 4a 4d, 5, and 6 are discussed in the respective Theorems The sequences on the curves labeled 4a 4d are defined in (4.5) and are discussed in the respective items i iv appearing after (4.8). This paper puts on a rigorous footing the idea, first introduced by Ginzburg and Landau, that low-order polynomial truncations of the free energy functional give correct asymptotic

6 Ellis, Machta, and Otto: Asymptotics for the Magnetization 6 results near continuous phase transitions for mean-field models [6]. The use of sequences (β n, K n ) that approach second-order points or the tricritical point permits us to establish the validity of truncating the expansion of the free-energy functional at an appropriate low order. The higher order terms are driven to zero by a power of n and are shown to be asymptotically irrelevant. While the renormalization group methodology also demonstrates the irrelevance of higher order terms in the expansion of the free-energy functional, it does so via a different route that depends on heuristics. By contrast, our approach is rigorous and shows in (.2) how to obtain the Ginzburg-Landau polynomial as a limit of suitably scaled free-energy functionals. No heuristic approximations or truncations appear. Let (β n, K n ) be any particular sequence converging to a second-order point or the tricritical point from the phase-coexistence region and denote the limiting point by (β, K). It is not difficult to obtain an asymptotic formula expressing the rate at which m(β n, K n ) converges to 0. Since m(β n, K n ) is the unique positive minimum point of G βn,k n, it solves the equation G β n,k n (m(β n, K n )) = 0. As we illustrate in appendix B of [2] in two examples, expanding G β n,k n in a Taylor series of appropriate order, one obtains the formula m(β n, K n ) x β β n γ for some x > 0 and γ > 0. However, this method gives only the functional form for x, not associating it with the model via the Ginzburg-Landau polynomial. This is in contrast to our general result in Theorem 3.2. That result identifies x as the unique positive, global minimum point of the Ginzburg-Landau polynomial, using the uniform convergence in (.2). The proof of that result completely avoids Taylor expansions, making use of the fact that under appropriate conditions, the positive global minimum points of n-dependent minimization problems converge to the positive global minimum point of a limiting minimization problem when such a minimum point is unique. Another contribution of the present paper is to demonstrate how the structure of the set of global minimum points of the associated Ginzburg-Landau polynomials g mirrors the phasetransition structure of the subsets through which (β n, K n ) passes. In this way the properties of the Ginzburg-Landau polynomials make rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena. Details of this mirroring are given in the discussions leading up to Theorem 4. and Theorem 4.4; of all the sequences that we consider, the sequence considered in the latter theorem shows the most varied behavior. Our work is also closely related to the scaling theory for critical and tricritical points. By choosing sequences that approach second-order points or the tricritical point from various directions and at various rates, we are able to verify a number of predictions of scaling theory. The sequences that approach the tricritical point reveal the subtle geometry of the crossover between critical and tricritical behavior described in Riedel s tricritical scaling theory [20]. In section 5 we will see that a proper application of scaling theory near the tricritical point requires that the scaling parameters be defined in a curvilinear coordinate system, an idea proposed in [20] but, to our knowledge, not previously explored.

7 Ellis, Machta, and Otto: Asymptotics for the Magnetization 7 Some of the results proved here can be obtained non-rigorously via the methods introduced by Capel and collaborators [5, 6, 7, 8, 7, 8, 9]. These papers introduce the mean-field B- C model and provide a general framework for studying mean-field models and obtaining the thermodynamic properties of these systems. In order to keep the present paper to a reasonable length, we have omitted a number of routine calculations. Full details can be found in our unpublished, companion paper [2]. We have also omitted the material in section 6 of that paper, in which properties of the Ginzburg- Landau polynomials, mathematical analysis, and numerical calculations are used to determine properties of the first-order curve in a neighborhood of the tricritical point. We end the introduction by previewing our results on the refined asymptotics of the total spin S N = N j= ω j, which are the main focus of the sequel to the present paper [3]. When N = n i.e., when the system size N coincides with the index n parametrizing the sequence (β n, K n ) these refined asymptotics reveal a fascinating relationship between the asymptotic formulas for m(β n, K n ) obtained here and the finite-size expectation S n /n n,βn,k n with respect to P n,βn,k n. For a wide class of sequences (β n, K n ) converging to a second-order point or the tricritical point, including the six sequences considered in the present paper, we prove that there exists a positive constant α 0 depending on the sequence and having the following properties. For α (0, α 0 ) the finite-size expectation S n /n n,βn,k n is asymptotic to m(β n, K n ). In this case (β n, K n ) converges slowly, and the system is in the phase-coexistence regime, where it is effectively infinite. On the other hand, when α > α 0, m(β n, K n ) is not related to the finite-size expectation. In this regime, the fluctuations of S n /n as measured by this expectation are much larger than m(β n, K n ), which converges to 0 at a much faster rate. When α > α 0, (β n, K n ) converges quickly, and the system is in the critical regime. The theory of finite-size scaling predicts that for such α, critical singularities are controlled by the size of the system rather than by the distance in parameter space from the phase transition [, 4, 9, 22]. The contents of the present paper are as follows. In section 2 we summarize the phasetransition structure of the mean-field B-C model. In section 3 we prove our main result (.) on the asymptotic behavior of m(β n, K n ) 0 [Thm. 3.2]. In section 4 that result is applied to six different sequences (β n, K n ), the first two of which converge to a second-order point and the last four of which converge to the tricritical point. In section 5 we relate the results obtained in the preceding section to the scaling theory of critical phenomena. In an appendix we state two results on polynomials of degree 6 needed in the paper. Acknowledgments. The research of Richard S. Ellis is supported in part by a grant from the National Science Foundation (NSF-DMS ).

8 Ellis, Machta, and Otto: Asymptotics for the Magnetization 8 2 Phase-Transition Structure of the Mean-Field B-C Model After defining the mean-field B-C model, we introduce a function G β,k, called the free-energy functional. The global minimum points of this function define the equilibrium values of the magnetization, and the minimum value of this function over R gives the canonical free energy. We then summarize the phase-transition structure of the model in terms of the behavior of the magnetization. This structure consists of a second-order phase transition, a first-order phase transition, and a tricritical point, which separates the two phase transitions. The mean-field B-C model is a lattice spin model defined on the complete graph on N vertices, 2,..., N. The spin at site j {, 2,..., N} is denoted by ω j, a quantity taking values in Λ = {, 0, }. The configuration space for the model is the set Λ n containing all sequences ω = (ω, ω 2,..., ω N ) with each ω j Λ. In terms of a positive parameter K representing the interaction strength, the Hamiltonian is defined by H N,K (ω) = ( N ωj 2 K N ) 2 ω j N j= j= for each ω Λ N. Let P N be the product measure on Λ N with identical one-dimensional marginals ρ = (δ 3 + δ 0 + δ ). Thus P N assigns the probability 3 N to each ω Λ N. For N N, inverse temperature β > 0, and K > 0, the canonical ensemble for the mean-field B-C model is the sequence of probability measures that assign to each subset B of Λ N the probability P N,β,K (B) = Z N (β, K) exp[ βh N,K (ω)] 3 N. ω B In this formula Z N (β, K) = ω Λ N exp[ βh N,K (ω)] 3 N. The analysis of the canonical ensemble P N,β,K is facilitated by absorbing the noninteracting component of the Hamiltonian into the product measure P N, obtaining P N,β,K (dω) = Z N (β, K) exp[ NβK(S N (ω)/n) 2] P N,β (dω). (2.) In this formula S N (ω) equals the total spin N j= ω j, P N,β is the product measure on Λ N with identical one-dimensional marginals ρ β (dω j ) = Z(β) exp( βω2 j )ρ(dω j), (2.2) Z(β) is the normalization equal to Λ exp( βω2 j)ρ(dω j ) = ( + 2e β )/3, and Z N (β, K) is the normalization equal to [Z(β)] N /Z N (β, K).

9 Ellis, Machta, and Otto: Asymptotics for the Magnetization 9 In order to summarize the phase-transition structure of the model, we introduce the cumulant generating function c β of ρ β, which for β > 0 and t R is defined by ( ) + e β (e t + e t ) c β (t) = log exp(tω )ρ β (dω ) = log. (2.3) + 2e β For β > 0, K > 0, and x R we also define Λ G β,k (x) = βkx 2 c β (2βKx). (2.4) The large deviation principle in Theorem 3.3 of [4] and the convexity analysis in Proposition 3.4 of [4] shows that if x is not a global minimum point of G β,k, then any sufficiently small interval containing x has an exponentially small probability with respect to the P N,β,K - distributions of S N /N; i.e., x is not observed in the canonical ensemble. Since the global minimum points of G β,k lie in [, ], we define the set M β,k of equilibrium values of the magnetization by M β,k = {x [, ] : x is a global minimum point of G β,k (x)}. (2.5) We call G β,k the free-energy functional of the mean-field B-C model because the canonical free energy ϕ(β, K), defined as lim N N log Z N (β, K), equals min x R G β,k (x). In section 2 of [2] the Ginzburg-Landau phenomenology is applied to G β,k in order to motivate the phase-transition structure of the model [6]. The next two theorems give the structure of M β,k first for 0 < β β c = log 4 and then for β > β c. These theorems make rigorous the predictions of the Ginzburg-Landau theory. The first theorem, proved in Theorem 3.6 in [4], describes the continuous bifurcation in M β,k for 0 < β β c. This bifurcation corresponds to a second-order phase transition. The quantity K(β) is denoted by K c (2) (β) in [4] and by K c (β) in [0]. Theorem 2.. For 0 < β β c, we define K(β) = /[2βc β(0)] = (e β + 2)/(4β). (2.6) For these values of β, M β,k has the following structure. (a) For 0 < K K(β), M β,k = {0}. (b) For K > K(β), there exists m(β, K) > 0 such that M β,k = {±m(β, K)}. (c) m(β, K) is a positive, increasing, continuous function for K > K(β), and as K (K(β)) +, m(β, K) 0 +. Therefore, M β,k exhibits a continuous bifurcation at K(β). The next theorem, proved in Theorem 3.8 in [4], describes the discontinuous bifurcation in M β,k for β > β c. This bifurcation corresponds to a first-order phase transition. As shown in that theorem, for all β > β c, K (β) < K(β). The quantity K (β) is denoted by K c () (β) in [4] and by K c (β) in [0].

10 Ellis, Machta, and Otto: Asymptotics for the Magnetization 0 Theorem 2.2. For β > β c, M β,k has the following structure in terms of the quantity K (β), denoted by K c () (β) in [4] and defined implicitly for β > β c on page 223 of [4]. (a) For 0 < K < K (β), M β,k = {0}. (b) For K = K (β) there exists m(β, K (β)) > 0 such that M β,k (β) = {0, ±m(β, K (β))}. (c) For K > K (β) there exists m(β, K) > 0 such that M β,k = {±m(β, K)}. (d) m(β, K) is a positive, increasing, continuous function for K K (β), and as K K (β) +, m(β, K) m(β, K (β)) > 0. Therefore, M β,k exhibits a discontinuous bifurcation at K (β). In the next section we present a general asymptotic result on the behavior of m(β n, K n ) for sequences (β n, K n ) converging either to a second-order point or to the tricritical point. In subsequent sections the theorem will be applied to a number of specific sequences. 3 Asymptotic Behavior of m(β n, K n ) in Terms of Ginzburg- Landau Polynomials Theorem 3.2 is the main result in this paper. It gives the asymptotic behavior of m(β n, K n ) for appropriate sequences (β n, K n ) lying in the phase-coexistence region and converging either to a second-order point or to the tricritical point. The asymptotic behavior is expressed in terms of the unique positive, global minimum point of the associated Ginzburg-Landau polynomial. The phase-coexistence region is defined to be all (β, K) satisfying 0 < β β c and K > K(β) and all (β, K) satisfying β > β c and K K (β). Thus for 0 < β β c, the phasecoexistence region consists of the region located above the second-order curve and above the tricritical point. For β > β c, the phase-coexistence region consists of the first-order curve (β, K (β)) and the region located above that curve. For all (β, K) in the phase-coexistence region there exists m(β, K) > 0 such that {±m(β, K)} M β,k. This is an equality for all (β, K) in the phase-coexistence region except for β > β c and K = K (β), in which case M β,k = {0, ±m(β, K)}. The first theorem in this section shows that for any sequence (β n, K n ) converging either to a second-order point or to the tricritical point, m(β n, K n ) 0. This theorem is an essential component in the proof of the asymptotic behavior of m(β n, K n ) given in Theorem 3.2. Theorem 3.. Let (β n, K n ) be an arbitrary positive sequence converging either to a secondorder point (β, K(β)), 0 < β < β c, or to the tricritical point (β, K(β)) = (β c, K(β c )). Then lim n m(β n, K n ) = 0.

11 Ellis, Machta, and Otto: Asymptotics for the Magnetization Proof. Since G βn,k n is a real analytic function, G βn,k n (m(β n, K n )) 0, and G βn,k n (x) as x, G βn,k n has a largest positive zero, which we denote by x n. We have the inequality 0 < m(β n, K n ) < x n. For any t R, c β (t) log(4e t ) = log 4 + t. Because the sequence (β n, K n ) is bounded and remains a positive distance from the origin and the coordinate axes, there exist numbers 0 < b < b 2 < such that b β n b 2 and b K n b 2 for all n N. Hence G βn,k n (x) = β n K n x 2 c βn (2β n K n x) β n K n x 2 2β n K n x log 4 b 2 ( x ) 2 b 2 2 log 4. Therefore, if x denotes the positive zero of the quadratic b 2 ( x ) 2 b 2 2 log 4, then 0 < sup n N m(β n, K n ) sup x n x. n N It follows that m(β n, K n ) is a bounded sequence. Thus given any subsequence m(β n, K n ), there exists a further subsequence m(β n2, K n2 ) and x R such that m(β n2, K n2 ) x as n 2. We complete the proof by showing that independently of the subsequence chosen, x = 0. To prove this, we use the fact that G βn2,k n2 (m(β n2, K n2 )) = inf y R G β n2,k n2 (y). Hence for any y R, G βn2,k n2 (m(β n2, K n2 )) G βn2,k n2 (y). Since G βn2,k n2 (x) G β,k(β) (x) uniformly for x in compact subsets of R, it follows that for all y R G β,k(β) ( x) = lim n 2 G β n2,k n2 (m(β n2, K n2 )) lim n 2 G β n2,k n2 (y) = G β,k(β) (y). Therefore x is a minimum point of G β,k(β). Because (β, K(β)) is either a second-order point or the tricritical point, x must coincide with the unique positive, global minimum point of G β,k(β) at 0 [Thm. 2.(a), Thm. 2.2(a)]. We have proved that any subsequence m(β n, K n ) of m(β n, K n ) has a further subsequence m(β n2, K n2 ) such that m(β n2, K n2 ) 0 as n 2. The conclusion is that lim n m(β n, K n ) = 0, as claimed. In section 4 we consider six different sequences (β n, K n ) converging either to a secondorder point or to the tricritical point. The fact that each of these sequences lies in the phasecoexistence region for all sufficiently large n is the first hypothesis of Theorem 3.2; this property implies that m(β n, K n ) > 0 for all sufficiently large n and m(β n, K n ) 0 [Thm. 3.]. Under three additional hypotheses Theorem 3.2 describes the exact asymptotic behavior of m(β n, K n ) 0. Examples of sequences for which the hypotheses of the theorem are valid are given in Theorems 4. and 4.2 for sequences converging to a second-order point and in Theorems for sequences converging to the tricritical point.

12 Ellis, Machta, and Otto: Asymptotics for the Magnetization 2 Theorem 3.2. Let (β n, K n ) be a positive sequence that converges either to a second-order point (β, K(β)), 0 < β < β c, or to the tricritical point (β, K(β)) = (β c, K(β c )). We assume that (β n, K n ) satisfies the following four hypotheses: (i) (β n, K n ) lies in the phase-coexistence region for all sufficiently large n. (ii) The sequence (β n, K n ) is parametrized by α > 0. This parameter regulates the speed of approach of (β n, K n ) to the second-order point or the tricritical point in the following sense: b = lim n α (β n β) and k = lim n α (K n K(β)) n n both exist, and b and k are not both 0; if b 0, then b equals or. (iii) There exists an even polynomial g of degree 4 or 6 satisfying g(x) as x together with the following two properties; g is called the Ginzburg-Landau polynomial. (a) α 0 > 0 and θ > 0 such that α > 0, if u = α/α 0 and γ = θα, then uniformly for x in compact subsets of R. lim n n u G βn,k n (x/n γ ) = g(x) (b) g has a unique positive, global minimum point x; thus the set of global minimum points of g equals {± x} or {0, ± x}. (iv) There exists a polynomial H satisfying H(x) as x together with the following property: α > 0 R > 0 such that, if u = α/α 0 and γ = θα, then n N sufficiently large and x R satisfying x/n γ < R, n u G βn,k n (x/n γ ) H(x). Under hypotheses (i) (iv), for any α > 0 m(β n, K n ) x/n θα ; i.e., lim n n θα m(β n, K n ) = x. If b 0, then this becomes m(β n, K n ) x β β n θ. Proof. Since G βn,k n (0) = 0 and G βn,k n is even, by hypotheses (iii) g is an even polynomial of degree 4 or 6 satisfying g(0) = 0. Hence the global minimum points of g are either ± x for some x > 0 or 0 and ± x for some x > 0. The proof of the asymptotic relationship m(β n, K n ) x/n θα is much easier in the case where the global minimum points of g are ± x for some x > 0. After a number of preliminary steps, we will prove the theorem for such polynomials g. We will then turn to the case where the global minimum points of g are 0 and ± x for some x > 0.

13 Ellis, Machta, and Otto: Asymptotics for the Magnetization 3 As in hypothesis (iii)(a), let α > 0 be given and define u = α/α 0 and γ = θα. In order to ease the notation, we write m n = n γ m(β n, K n ) and G n (x) = n u G βn,k n (x/n γ ). For all sufficiently large n, since (β n, K n ) lies in the phase-coexistence region, we have m(β n, K n ) > 0 and G βn,k n (m(β n, K n )) = inf y R G β n,k n (y). It follows that for all sufficiently large n G n ( m n ) = n u G βn,k n (m(β n, K n )) (3.) = inf y R [n u G βn,k n (y)] = inf y R G n(y); i.e., G n attains its minimum over R at m n > 0. This fact will be used several times in the proof. We first prove that the sequence { m n, n N} is bounded. If the sequence m n is not bounded, then there exists a subsequence m n of m n such that m n as n. Let R be the quantity in hypothesis (iv). Since m(β n, K n ) > 0 and m(β n, K n ) 0 [Thm. 3.], we have 0 < m n /n γ = m(β n, K n ) < R for all sufficiently large n, and so by hypothesis (iv) However, this contradicts the inequality G n ( m n ) H( m n ) as n. G n ( m n ) = inf y R G n(y) G n (0) = 0, which is valid for all n. This contradiction proves that the sequence m n is bounded. We now prove that m(β n, K n ) x/n γ = x/n θα in the case where the global minimum points of g are ± x for some x > 0. Let m n be any subsequence of m n. Since the sequence m n is bounded, there exists a further subsequence m n2 and ˆx 0 such that m n2 ˆx as n 2. According to (3.), for any y R, G n2 ( m n2 ) G n2 (y). Since G n (x) g(x) uniformly for x in compact subsets of R, it follows that g(ˆx) = lim n 2 G n 2 ( m n2 ) lim n 2 G n 2 (y) = g(y). Hence ˆx is a nonnegative global minimum point of g. Because g has a unique nonnegative, global minimum point x, which is positive, ˆx coincides with x. We have proved that any subsequence m n of m n has a further subsequence m n2 such that m n2 x as n 2. The conclusion is that lim n m n = x, which implies that m(β n, K n ) x/n γ = x/n θα. We now prove that m(β n, K n ) x/n γ = x/n θα in the case where the global minimum points of g are 0 and ± x for some x > 0. In this case g is a polynomial of degree 6. There are two subcases to consider: () there exists an infinite subsequence n in N such that the global

14 Ellis, Machta, and Otto: Asymptotics for the Magnetization 4 minimum points of G n are ± m n ; (2) there exists an infinite subsequence n 4 in N such that the global minimum points of G n4 are 0 and ± m n4. In subcase we will prove that any subsequence n 2 of n has a further subsequence n 3 for which m n3 x. This implies that m n x. In subcase 2 a similar proof shows that any subsequence n 5 of n 4 has a further subsequence n 6 for which m n6 x. This implies that m n4 x. Now let n 7 be an arbitrary subsequence in N. Then n 7 contains either infinitely many elements of the subsequence n or infinitely many elements of the subsequence n 4. In either case n 7 contains a further subsequence n 8 for which m n8 x. The conclusion is that m n x, which yields the desired conclusion, namely, m(β n, K n ) x/n γ = x/n θα. We focus on subcase ; subcase 2 is handled similarly. In order to understand the subtlety of the proof, we return to the argument just given in the case where the global minimum points of g are ± x for some x > 0. Let n be the subsequence in subcase and let n 2 be any further subsequence. Since the sequence m n2 is bounded, the same argument shows that there exists a further subsequence n 3 such that m n3 ˆx as n 2, where ˆx is a nonnegative global minimum point of g. When the global minimum points of g are ± x for some x > 0, we are able to conclude in fact that ˆx equals x. However, in the present case where the global minimum points of g are 0 and ± x for some x > 0, it might turn out that ˆx equals the global minimum point of g at 0. In this situation we would conclude that m n3 0, which is not the asymptotic relationship that we want. As this discussion shows, in subcase it suffices to prove that there exists no subsequence n 3 of n 2 for which m n3 0 as n 3. Under the assumption that there exists such a subsequence, we will reach a contradiction of the fact that m n3 is the largest critical point of G n3. This fact is not directly stated in [4], but it is a straightforward consequence of Lemmas 3.9 and 3.0(a) and Theorem 3.5 in that paper. From these three results it follows that when (β, K) lies in the phase-coexistence region, the positive global minimum point of the function F β,k (z) = G β,k (z/2βk) is also its largest critical point. From the definition of G n in terms of G βn,k n, it then follows that m n is the largest critical point of G n for all n. Since the global minimum points of g are 0 and ± x for some x > 0, there exists ȳ (0, x) such that g attains its maximum on the interval [0, x] at ȳ and attains its maximum on the interval [ x, 0] at ȳ. In addition, g(±ȳ) > 0 = g(0) = g(± x). The graph of g is shown in graph (a) in Figure 3. The graph of G n3 under the assumption that m n3 0 is shown in graph (b) in Figure 3. Referring to these graphs should help the reader follow the proof. By hypothesis (iii)(a), as n 3, G n3 (z) g(z) uniformly on compact subsets of R. Thus for all sufficiently large n 3 and each choice of sign G n3 (±ȳ) 2g(ȳ)/3 > 0, G n3 (± x) g(ȳ)/3. (3.2) By definition of subcase the global minimum points of G n3 are ± m n3, and by assumption m n3 0 as n 3. For all sufficiently large n 3, the inequality G n3 ( m n3 ) < G n3 (0) = 0

15 Ellis, Machta, and Otto: Asymptotics for the Magnetization 5 (a) g (b) G n 3 -M -x -y y x M -M -x -y y x M -mn mn 3 3 Figure 3: Proof of Theorem 3.2 in subcase. (a) Graph of Ginzburg-Landau polynomial g having three global minimum points, (b) graph of G n3 showing m n3 0. and the two inequalities in (3.2) imply that there exists ȳ n3 ( m n3, x) such that G n3 attains its maximum on the interval [ m n3, x] at ȳ n3 and attains its maximum on the interval [ x, m n3 ] at ȳ n3. Therefore, ȳ n3 is a critical point of G n3 greater than m n3. This contradicts the fact that m n3 is the largest critical point of G n3. The proof of subcase is complete. Subcase 2 is handled similarly. Let n 4 be the subsequence in subcase 2 and let n 5 be any further subsequence. If there exists a subsequence n 6 of n 5 for which m n6 0 as n 6, then for all sufficiently large n 6, there exists ȳ n6 ( m n6, x) such that G n6 attains its maximum on the interval [ m n6, x] at ȳ n6 and attains its maximum on the interval [ x, m n6 ] at ȳ n6. Again this contradicts the fact that m n6 is the largest critical point of G n6. This completes the proof of the theorem. In the next section we apply Theorem 3.2 to determine the asymptotic behavior of m(β n, K n ) for six sequences (β n, K n ) converging from the phase-coexistence region to a second-order point or to the tricritical point. 4 Asymptotic Behavior of m(β n, K n ) for Six Sequences In this section we derive the asymptotic behavior of the magnetization m(β n, K n ) for six sequences (β n, K n ). The first two sequences converge to a second-order point, and the last four sequences converge to the tricritical point. By definition, when (β n, K n ) lies in the phase-coexistence region, m(β n, K n ) is the unique

16 Ellis, Machta, and Otto: Asymptotics for the Magnetization 6 positive, global minimum point of the free-energy functional G βn,k n. For each of the sequences considered in this section the asymptotic behavior of m(β n, K n ) is expressed in terms of the unique positive, global minimum point x of the limit of a suitable scaled version of G βn,k n. This limit is a polynomial called the Ginzburg-Landau polynomial. As we will see, properties of this polynomial reflect the phase-transition structure of the mean-field B-C model in the region through which the associated sequence (β n, K n ) passes. This makes rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena discussed in more detail in section 2 of [2]. We first derive in Theorems 4. and 4.2 the asymptotic behavior of m(β n, K n ) for two sequences (β n, K n ) converging to a second-order point (β, K(β)) from the phase-coexistence region lying above the second-order curve. This asymptotic behavior is derived from the general result in Theorem 3.2. Full details of all the calculations for these two sequences are available in section 3 of [2]. For 0 < β < β c let (β n, K n ) be an arbitrary positive sequence converging to a second-order point (β, K(β)). According to hypothesis (iii) of Theorem 3.2, we seek numbers u R and γ > 0 and a suitable polynomial g such that n u G βn,k n (x/n γ ) g(x) uniformly on compact subsets of R. In order to carry this out, we consider the Taylor expansion for ng βn,k n (x/n γ ) to order 4 with an error term. Because G βn,k n (0) = 0, G βn,k n is an even function, and G (5) β,k (y) is uniformly bounded on compact subsets of [0, ) [0, ) R, Taylor s Theorem implies that for all n N, any R > 0, and all x R satisfying x/n γ < R ng βn,k n (x/n γ ) = (2) G n 2γ 2! β n,k n (0) x 2 + (4) G n 4γ 4! β n,k n (0) ( ) x 4 + O x 5. (4.) n 5γ In this formula the big-oh term is uniform for x ( Rn γ, Rn γ ). Define c 4 (β) = (e β + 2) 2 (4 e β )/(8 4!), which is positive since 0 < β < β c = log 4, and let ε n denote a sequence that converges to 0 and that represents the various error terms arising in the following calculation. If we substitute into the last display the formulas for G (2) β n,k n (0) and G (4) β n,k n (0) and use the convergence (β n, K n ) (β, K(β)) and the continuity of K( ), then the last display implies that for all n N, any u R, any γ > 0, any R > 0, and all x R satisfying x/n γ < R n u G βn,k n (x/n γ ) = n 2γ +uβ(k(β n) K n )( + ε n )x 2 (4.2) + ( ) n 4γ +uc 4(β)( + ε n )x 4 + O x 5, n 5γ where the big-oh term is uniform for x ( Rn γ, Rn γ ).

17 Ellis, Machta, and Otto: Asymptotics for the Magnetization 7 The two different asymptotic behaviors of m(β n, K n ) to be considered first in this section each depends on the choice of the sequence (β n, K n ) converging to the second-order point (β, K(β)). Each choice controls, in a different way, the rate at which (K(β n ) K n ) in the quadratic term in (4.2) converges to 0. Fix 0 < β < β c. For the first choice of sequence we take α > 0, b {, 0, }, and a real number k K (β)b and define β n = β + b/n α and K n = K(β) + k/n α. (4.3) Since K(β n ) = K(β + b/n α ) = K(β) + K (β)b/n α + O(/n 2α ), it follows from (4.2) that for all n N, any u R, any γ > 0, any R > 0, and all x R satisfying x/n γ < R n u G βn,k n (x/n γ ) (4.4) = (β)b k)( + ε n 2γ+α +uβ(k n )x 2 ( ) ( ) + n 4γ +uc 4(β)( + ε n )x 4 + O x 2 + O x 5. n 2γ+2α +u n 5γ +u We now impose the condition that the powers of n appearing in the first two terms of the last display equal 0; i.e., 2γ + α + u = 0 = 4γ + u. These two equalities are equivalent to γ = α/2 and u = 4γ = 2α; in the notation of hypothesis (iii)(a) of Theorem 3.2 u = α/α 0 and γ = θα, where α 0 = /2 and θ = /2. With this choice of γ and u the powers of n appearing in the last two terms in (4.4) are positive. It follows that as n, we have uniformly for x in compact subsets of R lim n n u G βn,k n (x/n γ ) = g(x) = β(k (β)b k)x 2 + c 4 (β)x 4. (4.5) We now further assume that K (β)b k < 0. This inequality implies that (β n, K n ) converges to (β, K(β)) along a ray lying above the tangent line to (β, K(β)) (see path in Figure 2). Thus, in accordance with hypothesis (i) of Theorem 3.2 (β n, K n ) lies in the phase-coexistence region above the second-order curve for all sufficiently large n. Since b and k are not both 0, hypothesis (ii) of that theorem is also valid. As required by hypothesis (iii) of Theorem 3.2, the Ginzburg-Landau polynomial g is an even polynomial of degree 4, and since K (β)b k < 0 and c 4 (β) > 0, g(x) as x and g has a unique positive, global minimum point. Substituting γ = α/2 and u = 2α into (4.4), one proves the lower bound in hypothesis (iv) of Theorem 3.2 with H(x) = 2β K (β)b k x c 4(β)x 4. For all n for which (β n, K n ) lies in the phase-coexistence region, the global minimum points of the free energy functional G βn,k n are ±m(β n, K n ) [Thm. 2.(b)], a property reflected in the fact that the global minimum points of the Ginzburg-Landau polynomial g are also a pair of

18 Ellis, Machta, and Otto: Asymptotics for the Magnetization 8 symmetric nonzero points. Alternatively, if K (β)b k > 0, then (β n, K n ) lies in the singlephase region under the second-order curve for all sufficiently large n, G βn,k n has a unique global minimum point at 0 [Thm. 2.(a)], and g has a unique global minimum point at 0. In this way properties of g reflect the phase-transition structure of the model in the region through which (β n, K n ) passes. In the next theorem we describe the asymptotic behavior of m(β n, K n ) for the sequence (β n, K n ) defined in (4.3) when K (β)b k < 0. Since θ = /2, Theorem 3.2 implies that m(β n, K n ) x/n θα = x/n α/2 0, where x denotes the unique positive, global minimum point of g. The next theorem, the proof of which has just been sketched, corresponds to Theorem 3. in [2], where full details are given. Theorem 4.. For β (0, β c ), α > 0, b {, 0, }, and a real number k K (β)b, define β n = β + b/n α and K n = K(β) + k/n α as well as c 4 (β) = (e β + 2) 2 (4 e β )/(8 4!). Then (β n, K n ) converges to the second-order point (β, K(β)). The following conclusions hold. (a) For any α > 0, u = 2α, and γ = α/2 lim n n u G βn,k n (x/n γ ) = g(x) = β(k (β)b k)x 2 + c 4 (β)x 4 uniformly for x in compact subsets of R. (b) The Ginzburg-Landau polynomial g has nonzero global minimum points if and only if K (β)b k < 0. If this inequality holds, then the global minimum points of g are ± x, where x = (β(k K (β)b)/[2c 4 (β)]) /2 (4.6) (c) Assume that k > K (β)b. Then for any α > 0, m(β n, K n ) 0 and has the asymptotic behavior m(β n, K n ) x/n α/2 ; i.e., lim n n α/2 m(β n, K n ) = x. If b 0, then this becomes m(β n, K n ) x β β n /2. We now consider the second choice of the sequence (β n, K n ) converging to a second-order point (β, K(β)) corresponding to 0 < β < β c. Given α > 0, b {, }, an integer p 2, and a real number l K (p) (β), we define p β n = β + b/n α and K n = K(β) + K (j) (β)b j /(j!n jα ) + lb p /(p!n pα ). (4.7) j=

19 Ellis, Machta, and Otto: Asymptotics for the Magnetization 9 This sequence converges to the second-order point along a curve that coincides with the secondorder curve to order n (p )α (see path 2 in Figure 2). Since the expansion (4.2) takes the form K(β n ) K n = (K (p) (β) l)b p /(p!n pα ) + O(/n (p+)α) ), n u G βn,k n (x/n γ ) (4.8) = n 2γ+pα +u p! β(k(p) (β) l)b p ( + ε n )x 2 + ( ) ( ) n 4γ +uc 4(β)( + ε n )x 4 + O x 2 + O x 5. n 2γ+(p+)α +u n 5γ +u This formula is valid for all n N, any u R, any γ > 0, any R > 0, and all x R satisfying x/n γ < R, and ε n 0. We now impose the condition that the powers of n in the first two terms of the last display equal 0; i.e., 2γ + pα + u = 0 = 4γ + u. These two equalities are equivalent to γ = pα/2 and u = 4γ = 2pα; in the notation of hypothesis (iii)(a) of Theorem 3.2 u = α/α 0 and γ = θα, where α 0 = /(2p) and θ = p/2. With this choice of γ and u, the powers of n in the last two terms in (4.8) are positive. It follows that as n, we have uniformly for x in compact subsets of R lim n n u G βn,k n (x/n γ ) = g(x) = p! β(k(p) (β) l)b p x 2 + c 4 (β)x 4. We now further assume that (K (p) (β) l)b p < 0. This inequality implies that K n > K(β n ) for all sufficiently large n. Hence in accordance with hypothesis (i) of Theorem 3.2, (β n, K n ) lies in the phase-coexistence region above the second-order curve for all sufficiently large n. Since b 0, hypothesis (ii) of that theorem is also valid. As required by hypothesis (iii) of Theorem 3.2, the Ginzburg-Landau polynomial g is an even polynomial of degree 4, and since (K (p) (β) l)b p < 0 and c 4 (β) > 0, g(x) as x and g has a unique positive, global minimum point. As in the case of sequence (4.3), properties of g reflect the phasetransition structure of the model in the region through which (β n, K n ) passes. Substituting γ = pα/2 and u = 2pα into (4.8), one verifies hypothesis (iv) of Theorem 3.2 with H(x) = 2 p! β K(p) (β)b l x c 4(β)x 4. This completes the verification of the hypotheses of Theorem 3.2. In the next theorem we describe the asymptotic behavior of m(β n, K n ) for the sequence (β n, K n ) defined in (4.7) when (K (p) (β) l)b p < 0. Since θ = p/2, Theorem 3.2 implies that m(β n, K n ) x/n θα = x/n pα/2 0, where x denotes the unique positive, global minimum point of g. The next theorem, the proof of which has just been sketched, corresponds to Theorem 3.2 in [2], where full details are given.

20 Ellis, Machta, and Otto: Asymptotics for the Magnetization 20 Theorem 4.2. For β (0, β c ), α > 0, b {, }, an integer p 2, and a real number l K (p) (β), define p β n = β + b/n α and K n = K(β) + K (j) (β)b j /(j!n jα ) + lb p /(p!n pα ) as well as c 4 (β) = (e β + 2) 2 (4 e β )/(8 4!). Then (β n, K n ) converges to the second-order point (β, K(β)). The following conclusions hold. (a) For any α > 0, u = 2pα, and γ = pα/2 lim n n u G βn,k n (x/n γ ) = g(x) = p! β(k(p) (β) l)b p x 2 + c 4 (β)x 4 uniformly for x in compact subsets of R. (b) The Ginzburg-Landau polynomial g has nonzero global minimum points if and only if (K (p) (β) l)b p < 0. If this inequality holds, then the global minimum points of g are ± x, where x = ( β(l K (p) (β))b p /[2c 4 (β)p!] ) /2. (4.9) (c) Assume that (K (p) (β) l)b p < 0. Then for any α > 0, m(β n, K n ) 0 and has the asymptotic behavior j= m(β n, K n ) x/n pα/2 = x β β n p/2 ; i.e., lim n n pα/2 m(β n, K n ) = x. We next derive in Theorems the asymptotic behavior of m(β n, K n ) for four sequences (β n, K n ) converging to the tricritical point (β c, K(β c )) from various subsets of the phase-coexistence region. A number of new phenomena arise in this case that are not observed in the cases considered earlier in this section. Full details of all the calculations for these four sequences are available in section 5 of [2]. As in the case of the two sequences considered earlier in this section, properties of the Ginzburg-Landau polynomials for these four new sequences reflect the phase-transition structure of the mean-field B-C model in the region through which the associated sequence (β n, K n ) passes. This again makes rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena discussed in section 2 of [2]. Let (β n, K n ) be an arbitrary positive sequence converging to the tricritical point (β c, K(β c )) = (log 4, 3/2log 4). According to hypothesis (iii) of Theorem 3.2, we seek numbers u R and γ R and a suitable polynomial g such that n u G βn,k n (x/n γ ) g(x) uniformly on compact subsets of R. In order to carry this out, we consider the Taylor expansion of ng βn,k n (x/n γ ) to

21 Ellis, Machta, and Otto: Asymptotics for the Magnetization 2 order 6 with an error term. This expansion takes the form ng βn,k n (x/n γ ) = G (2) n 2γ 2! + (6) G n 6γ 6! β n,k n (0) β n,k n (0) x 2 + G (4) β n,k n (0) x 4 (4.0) n 4γ 4! ( ) x 6 + O x 7, n 7γ which is valid for all n N, any R > 0, and all x R satisfying x/n γ < R. The big-oh term is uniform for x ( Rn γ, Rn γ ). Define c 4 = 3/6 and c 6 = 9/40 and let ε n denote a sequence that converges to 0 and that represents the various error terms arising in the following calculation. If we substitute into the last display the formulas for G (2) β n,k n (0), G (4) β n,k n (0), and G (6) β n,k n (0) and use the convergence (β n, K n ) (β c, K(β c )) and the continuity of K( ), the last display implies that for all n N, any u R, any γ > 0, and all x R satisfying x/n γ < R n u G βn,k n (x/n γ ) (4.) = n 2γ +uβ c(k(β n ) K n )( + ε n )x 2 + n 4γ +uc 4(4 e βn )( + ε n )x 4 ( ) + n 6γ +uc 6( + ε n )x 6 + O x 7. n 7γ +u The four different asymptotic behaviors of m(β n, K n ) to be considered in the remainder of this section each depends on the choice of the sequence (β n, K n ) converging to the tricritical point (β c, K(β c )). Each choice controls, in a different way, the rate at which (K(β n ) K n ) in the quadratic term in (4.) and the rate at which (4 e βn ) in the quartic term converge to 0. For the first choice of sequence we take α > 0, b {, 0, }, and a real number k K (β c )b and define β n = β c + b/n α and K n = K(β c ) + k/n α. (4.2) Since K(β n ) K n = (K (β c )b k)/n α + O(/n 2α ) and 4 e βn = 4b/n α + O(/n 2α ), it follows from (4.) that for all n N, any u R, any γ > 0, any R > 0, and all x R satisfying x/n γ < R n u G βn,k n (x/n γ ) (4.3) = n 2γ+α +uβ c(k (β c )b k)( + ε n )x 2 n 4γ+α +u4c 4b( + ε n )x 4 ( ) + n 6γ +uc 6( + ε n )x 6 + O x 2 n ( ) ( 2γ+2α +u ) + O x 4 + O x 7. n 4γ+2α +u n 7γ +u

22 Ellis, Machta, and Otto: Asymptotics for the Magnetization 22 We now impose the condition that the powers of n appearing in the first and third terms in the last display equal 0; i.e., 2γ + α +u = 0 = 6γ +u. These two equalities are equivalent to γ = α/4 and u = 6γ = 3α/2; in the notation of hypothesis (iii)(a) of Theorem 3.2 u = α/α 0 and γ = θα, where α 0 = 2/3 and θ = /4. With this choice of γ and u, the powers of n in the second term and the last three terms in (4.3) are positive. It follows that as n, we have for uniformly for x in compact subsets of R lim n n u G βn,k n (x/n γ ) = g(x) = β c (K (β)b k)x 2 + c 6 x 6. We now further assume that k > K (β c )b. This inequality implies that (β n, K n ) converges to (β c, K(β c )) along a ray lying above the tangent line to (β c, K(β c )) (see path 3 in Figure 2). Hence in accordance with hypothesis (i) of Theorem 3.2, (β n, K n ) lies in the phase-coexistence region for all sufficiently large n; (β n, K n ) is above the spinodal curve if b = (see Figure ), above the second-order curve if b =, and above the tricritical point if b = 0. Since b and k are not both 0, hypothesis (ii) of Theorem 3.2 is also valid. As required by hypothesis (iii) of Theorem 3.2, the Ginzburg-Landau polynomial g is an even polynomial of degree 6, and since K (β c )b k < 0 and c 6 > 0, g(x) as x and g has a unique positive, global minimum point. As in the case of the sequence in (4.3), properties of g reflect the phasetransition structure of the model in the region through which (β n, K n ) passes. Substituting γ = α/4 and u = 3α/2 into (4.3), one verifies hypothesis (iv) of Theorem 3.2 with H(x) = 2β c K (β c )b k x 2 8c 4 x c 6x 6. This completes the verification of the hypotheses of Theorem 3.2. In the next theorem we describe the asymptotic behavior of m(β n, K n ) for the sequence (β n, K n ) defined in (4.2) when K (β c )b k < 0. Since θ = /4, Theorem 3.2 implies that m(β n, K n ) x/n θα = x/n α/4 0, where x denotes the unique positive, global minimum point of g. The next theorem, the proof of which has just been sketched, corresponds to Theorem 5. in [2], where full details are given. Theorem 4.3. For α > 0, b {, 0, }, and a real number k K (β c )b, define β n = β c + b/n α and K n = K(β c ) + k/n α as well as c 6 = 9/40. Then (β n, K n ) converges to the tricritical point (β c, K(β c )). The following conclusions hold. (a) For any α > 0, u = 3α/2, and γ = α/4 lim n n u G βn,k n (x/n γ ) = g(x) = β c (K (β c )b k)x 2 + c 6 x 6 uniformly for x in compact subsets of R.