Asymptotic behavior of the finitesize magnetization as a function of the speed of approach to criticality
Abstract
The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finitesize magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a secondorder point or the tricritical point in the meanfield Blume–Capel model. We show that the thermodynamic magnetization and the finitesize magnetization are asymptotic when the parameter governing the speed at which the sequence approaches criticality is below a certain threshold . However, when exceeds , the thermodynamic magnetization converges to 0 much faster than the finitesize magnetization. The asymptotic behavior of the finitesize magnetization is proved via a moderate deviation principle when and via a weakconvergence limit when . To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finitesize scaling for a meanfield model.
10.1214/10AAP679 \volume20 \issue6 2010 \firstpage2118 \lastpage2161
Asymptotic behavior of the finitesize magnetization
A]\fnmsRichard S. \snmEllis\corref\thanksreft1label=e1]rsellis@math.umass.edu, B]\fnmsJonathan \snmMachtalabel=e2]machta@physics.umass.edu and C]\fnmsPeter TakHun \snmOttolabel=e3]potto@willamette.edu
t1Supported by NSF Grant DMS0604071.
class=AMS] \kwd[Primary ]60F10 \kwd60F05 \kwd[; secondary ]82B20. Finitesize magnetization \kwdthermodynamic magnetization \kwdsecondorder phase transition \kwdfirstorder phase transition \kwdtricritical point \kwdmoderate deviation principle \kwdlarge deviation principle \kwdscaling limit \kwdBlume–Capel model \kwdfinitesize scaling.
1 Introduction
For the meanfield Blume–Capel model, as for other meanfield spin systems, the magnetization in the thermodynamic limit is well understood within the theory of large deviations. In this framework the thermodynamic magnetization arises as the unique, positive, global minimum point of the rate function in a large deviation principle. The question answered in this paper is whether, in a neighborhood of criticality, the thermodynamic magnetization is a physically relevant estimator of the finitesize magnetization, which is the expected value of the spin per site. A similar question is answered by the heuristic, statistical mechanical theory of finitesize scaling. This paper is both motivated by the theory of finitesize scaling and puts that theory on a firm foundation in the context of meanfield spin systems. It is hoped that our results suggest how this question can be addressed in the context of much more complicated, shortrange spin systems.
Our approach is to evaluate the asymptotic behaviors of the thermodynamic magnetization and the physically relevant, finitesize magnetization along parameter sequences converging to either a secondorder point or the tricritical point in the meanfield Blume–Capel model. The thermodynamic magnetization is then considered to be a physically relevant estimator of the finitesize magnetization when these two quantities have the same asymptotic behavior. Our main finding is that the value of the parameter governing the speed at which the sequence approaches criticality determines whether or not the asymptotic behaviors of these two quantities are the same. Specifically, we show in Theorem 4.1 that the thermodynamic magnetization and the finitesize magnetization are asymptotic when is below a certain threshold and that therefore the thermodynamic magnetization is a physically relevant estimator when . However, when exceeds , then according to Theorem 4.2, the thermodynamic magnetization converges to 0 much faster than the finitesize magnetization, and therefore the thermodynamic magnetization is not a physically relevant estimator when . An advantage of using the thermodynamic magnetization as an estimator of the finitestate magnetization when is that the asymptotic behavior of the former quantity is much easier to derive than the asymptotic behavior of the latter quantity [see the discussion at the end of the paragraph after (5)].
The investigation is carried out for a meanfield version of an important lattice spin model due to Blume and Capel, to which we refer as the B–C model [4, 6, 7, 8]. This meanfield model is equivalent to the B–C model on the complete graph on vertices. It is one of the simplest models that exhibits the following intricate phasetransition structure: a curve of secondorder points, a curve of firstorder points and a tricritical point, which separates the two curves. A generalization of the B–C model is studied in [5].
The meanfield B–C model is defined by a canonical ensemble that we denote by ; equals the number of spins, is the inverse temperature and is the interaction strength. is defined in (7) in terms of the Hamiltonian
in which represents the spin at site and takes values in . The configuration space for the model is the set containing all sequences with each . Expectation with respect to is denoted by . The finitesize magnetization is defined by , where equals the total spin .
Before introducing the results in this paper, we summarize the phasetransition structure of the model. For and we denote by the set of equilibrium values of the magnetization. coincides with the set of global minimum points of the freeenergy functional , which is defined in (11). It is known from heuristic arguments and is proved in [16] that there exists a critical inverse temperature and that for there exists a quantity and for there exists a quantity having the following properties. The positive quantity appearing in the following list is the thermodynamic magnetization.

Fix . Then for , consists of the unique pure phase 0, and for , consists of two nonzero values .

For , undergoes a continuous bifurcation at , changing continuously from for to for . This continuous bifurcation corresponds to a secondorder phase transition.

Fix . Then for , consists of the unique pure phase 0, for , consists of 0 and two nonzero values and for , consists of two nonzero values .

For , undergoes a discontinuous bifurcation at , changing discontinuously from for to for to for . This discontinuous bifurcation corresponds to a firstorder phase transition.
Because of items 2 and 4, we refer to the curve as the secondorder curve and to the curve as the firstorder curve. Points on the secondorder curve are called secondorder points, and points on the firstorder curve firstorder points. The point separates the secondorder curve from the firstorder curve and is called the tricritical point. The twophase region consists of all points in the positive  quadrant for which consists of two values. Thus this region consists of all above the secondorder curve, above the tricritical point and above the firstorder curve; that is, all satisfying and and satisfying and . The sets that describe the phasetransition structure of the model are shown in Figure 1.
For fixed lying in the twophase region the finitesize magnetization converges to the thermodynamic magnetization as . In order to see this, we use the large deviation principle (LDP) for with respect to in [16], Theorem 3.3, and the fact that the set of global minimum points of the rate function in that LDP coincides with the set [16], Proposition 3.4, the structure of which has just been described. Since for lying in the twophase region , the LDP implies that the distributions of put an exponentially small mass on the complement of any open set containing . Symmetry then yields the weakconvergence limit
(1) 
This implies the desired result
(2) 
The limit in the last display is closely related to the main focus of this paper. It shows that because the thermodynamic magnetization is the limit, as the number of spins goes to , of the finitesize magnetization, the thermodynamic magnetization is a physical relevant estimator of the finitesize magnetization, at least when evaluated at fixed in the twophase region.
The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finitesize magnetization in a more general sense, namely, when evaluated along a class of sequences that converge to a secondorder point or the tricritical point . The criterion for determining whether is a physically relevant estimator is that as , is asymptotic to the finitesize magnetization , both of which converge to 0. In this formulation we let in the finitesize magnetization; that is, we let the number of spins coincide with the index parametrizing the sequence . As summarized in Theorems 4.1 and 4.2, our main finding is that is a physically relevant estimator if the parameter governing the speed at which approaches criticality is below a certain threshold ; however, this is not true if . For the sequences under consideration the parameter determines the limits
which are assumed to exist and not to be both 0. The value of depends on the type of the phase transition—firstorder, secondorder or tricritical—that influences the sequence, an issue addressed in Section 5 of [13].
We illustrate the results contained in these two theorems by applying them to six types of sequences. In the case of secondorder points two such sequences are considered in Theorems 5.1 and 5.2, and in the case of the tricritical point four such sequences are considered in Theorems 5.3–5.6. Possible paths followed by these sequences are shown in Figure 2. We believe that modulo uninteresting scale changes, irrelevant higher order terms and other inconsequential modifications, these are all the sequences of the form and equal to plus a polynomial in , where is either a secondorder point or the tricritical point and for some and .
We next summarize our main results on the asymptotic behaviors of the thermodynamic magnetization and the finitesize magnetization, first for small values of and then for large values of . The relevant information is given, respectively, in Theorems 3.1, 4.1 and 4.2. These theorems are valid for suitable positive sequences parametrized by , lying in the twophase region for all sufficiently large , and converging either to a secondorder point or to the tricritical point. The hypotheses of these three theorems overlap but do not coincide. The hypotheses of Theorem 3.1 are satisfied by all six sequences considered in Section 5 while the hypotheses of each of the Theorems 4.1 and 4.2 are satisfied by all six sequences with one exception. For each of the six sequences the quantities and appearing in these asymptotic results are specified in Table 1.
Seq.  Defn.  Thm.  

1  (14)  Theorem 5.1  
2  (16)  Theorem 5.2  
3  (18)  Theorem 5.3  
4  (19)  Theorem 5.4  
5  (21)  Theorem 5.5  
6  (23)  Theorem 5.6 
The difference in the asymptotic behaviors of the thermodynamic magnetization and the finitesize magnetization for is described in item 3. As we discuss in Section 6, the difference is explained by the statistical mechanical theory of finitesize scaling.

According to Theorem 3.1, there exists positive quantities and such that for all
(3) 
(). According to Theorem 4.1, there exists a threshold value such that for all
(4) Because is asymptotic to the finitesize magnetization, is a physically relevant estimator of the finitesize magnetization. In this case converges to criticality slowly, and we are in the twophase region, where the system is effectively infinite. Formally the first index parametrizing the finitesize magnetization can be sent to before the index parametrizing the sequence is sent to , and so we have

(). According to Theorem 4.2, there exists a positive quantity such that for all
(5) Because converges to 0 much faster than the finitesize magnetization, is not a physically relevant estimator of the finitesize magnetization. In this case converges to criticality quickly, and we are in the critical regime, where finitesize scaling effects are important.
The asymptotic behavior of the thermodynamic magnetization stated in (3) holds for all . It is derived in Theorem 3.2 in [13] and is summarized in Theorem 3.1 in the present paper. In (4) we state the asymptotic behavior of the finitesize magnetization for . This result is proved in part (a) of Theorem 4.1 as a consequence of the moderate deviation principle (MDP) for the spin in Theorem 7.1, the weakconvergence limit in Corollary 7.3, and the uniform integrability estimate in Lemma 7.4. The asymptotic behavior of stated in (5) for is proved in part (a) of Theorem 4.2 as a consequence of the weakconvergence limit for the spin in Theorem 8.1 and the uniformintegrabilitytype estimate in Proposition 8.3. In part (a) of Theorem 4.3 we state the asymptotic behavior of for . That result is a consequence of a weakconvergence limit analogous to the limit in Theorem 8.1 and the uniformintegrabilitytype estimate in Proposition 8.3. With changes in notation only, Theorem 3.1 and Theorems 4.1–4.3 also apply to other meanfield models including the Curie–Weiss model [12] and the Curie–Weiss–Potts model [17]. The proof of the asymptotic behavior of the thermodynamic magnetization in [13], Theorem 3.2, is purely analytic and is much more straightforward than the probabilistic proofs of the asymptotic behaviors of the finitesize magnetization in Theorems 4.1–4.3.
Figure 3 gives a pictorial representation of the phenomena that are summarized in (4) for and in (5) for . As we discuss in Section 2, for the sequences under consideration the thermodynamic magnetization can be characterized as the unique, positive, global minimum point in an LDP or, equivalently, as the unique, positive, global minimum point of the dual, freeenergy functional defined in (11). According to graph (a) in Figure 3, for , has two deep, global minimum points at . Graph (b) in Figure 3, which is not shown to scale, exhibits the contrasting situation for . In this case the global minimum points of at are shallow and close to the origin. In the two graphs we also show the form of the distribution . For this probability distribution is sharply peaked at as . In contrast, for the probability distribution is peaked at 0 and its standard deviation is much larger than .
In a work in progress we refine the asymptotic result in (4), which states that for , is asymptotic to as . Define , which exceeds since . We conjecture that for a class of suitable sequences that includes the first five sequences considered in Section 5, there exists a positive quantity such that for all
(6) 
This refined asymptotic result would extend part (b) of Theorem 4.1. It is a consequence of the conjecture that when is conditioned to lie in a suitable neighborhood of , the distributions of converge in distribution to a Gaussian.
For easy reference we list in Table 1 information about the six sequences considered in Section 5. The first two columns list, respectively, the equation in which each sequence is defined and the theorem in which the asymptotic results in equations (3), (4) and (5) are stated for each sequence. In these theorems the quantities and appearing in the three asymptotic results are defined. The three asymptotic results involve the quantities , , and , the values of the first two of which are listed in the next two columns of the table. In the last column of the table we list the values of . Through the factor , governs the conjectured asymptotics of stated in (6).
The contents of this paper are as follows. In Section 2 we summarize the phasetransition structure of the meanfield B–C model. Theorem 3.1 in Section 3 gives the asymptotic behavior of the thermodynamic magnetization for suitable sequences converging either to a secondorder point or to the tricritical point. The heart of the paper is Section 4. In this section Theorems 4.1, 4.2 and 4.3 give the asymptotic behavior of the finitesize magnetization for three respective ranges of , and . The quantity is a threshold value that depends on the type of the phase transition—firstorder, secondorder or tricritical—that influences the associated sequence . These theorems also compare the asymptotic behaviors of the thermodynamic magnetization and the finitesize magnetization, showing that they are the same for but not the same for . In Section 5 the three theorems in the preceding section are applied to six specific sequences , the first two of which converge to a secondorder point and the last four of which converge to the tricritical point. Section 6 gives an overview of the statistical mechanical theory of finitesize scaling, which gives insight into the physical phenomena underlying our mathematical results. Part (a) of Theorem 4.1 is derived in Section 7 from the MDP for the spin in Theorem 7.1, the weakconvergence limit for the spin in Corollary 7.3, and the uniform integrability estimate in Lemma 7.4. Finally, part (a) of Theorem 4.2 is derived in Section 8 from the weakconvergence limit for the spin in Theorem 8.1 and the uniformintegrabilitytype estimate in Proposition 8.3.
2 Phasetransition structure of the meanfield B–C model
After defining the meanfield B–C model, we introduce a function , called the freeenergy functional. The global minimum points of this function define the equilibrium values of the magnetization. The phasetransition structure of the model is summarized in Theorems 2.1 and 2.2. The first theorem shows that the model exhibits a secondorder phase transition for , where is the critical inverse temperature of the model. The second theorem shows that the model exhibits a firstorder phase transition for .
For the meanfield B–C model is a latticespin model defined on the complete graph on vertices . The spin at site is denoted by , a quantity taking values in . The configuration space for the model is the set containing all sequences with each . In terms of a positive parameter representing the interaction strength, the Hamiltonian is defined by
for each . Let be the product measure on with identical onedimensional marginals . Thus assigns the probability to each . For inverse temperature and for , the canonical ensemble for the meanfield B–C model is the sequence of probability measures that assign to each subset of the probability
(7)  
In this formula is the partition function equal to
Expectation with respect to is denoted by .
The analysis of the canonical ensemble is facilitated by absorbing the noninteracting component of the Hamiltonian into the product measure , obtaining
(8) 
In this formula equals the total spin , is the product measure on with identical onedimensional marginals
(9) 
is the normalization equal to and is the normalization equal to .
We denote by the set of equilibrium macrostates of the meanfield B–C model. In order to describe this set, we introduce the cumulant generating function of the measure defined in (9); for this function is defined by
(10)  
For we define
(11) 
As shown in Proposition 3.4 in [16], the set of equilibrium macrostates of the meanfield B–C model can be characterized as the set of global minimum points of :
(12) 
In [16] the set was denoted by .
We also define the canonical free energy
where is the normalizing constant in (8). This limit exists and equals . Because of this property of , we call the freeenergy functional of the meanfield B–C model.
The next two theorems use (12) to determine the structure of for and for . The positive quantity appearing in these theorems is called the thermodynamic magnetization. The first theorem, proved in Theorem 3.6 in [16], describes the continuous bifurcation in for as crosses a curve . This bifurcation corresponds to a secondorder phase transition, and this curve is called the secondorder curve. The quantity , defined in (13), is denoted by in [16].
Theorem 2.1
For , we define
(13) 
For these values of , has the following structure:
(a) For , .
(b) For , there exists such that .
(c) is a positive, increasing, continuous function for , and as , . Therefore, exhibits a continuous bifurcation at .
The next theorem, proved in Theorem 3.8 in [16], describes the discontinuous bifurcation in for as crosses a curve . This bifurcation corresponds to a firstorder phase transition, and this curve is called the firstorder curve. As shown in Theorem 3.8 in [16], for all , . The quantity is denoted by in [16].
Theorem 2.2
For , has the following structure in terms of the quantity , denoted by in [16] and defined implicitly for on page 2231 of [16]:
(a) For , .
(b) For there exists such that .
(c) For there exists such that .
(d) is a positive, increasing, continuous function for , and as , . Therefore, exhibits a discontinuous bifurcation at .
The phasecoexistence region is defined as the set of all points in the positive  quadrant for which consists of more than one value. According to Theorems 2.1 and 2.2, the phasecoexistence region consists of all points above the secondorder curve, above the tricritical point, on the firstorder curve and above the firstorder curve; that is,
Our derivation of the asymptotic behavior of the finitesize magnetization in this paper is valid for a class of sequences lying in the phasecoexistence region for all sufficiently large and converging either to a secondorder point or to the tricritical point. In the next section we state an asymptotic formula for for a general class of such sequences. That asymptotic formula will be used later in the paper when we study the asymptotic behavior of the finitesize magnetization .
3 Asymptotic behavior of
The main result in this section is Theorem 3.1. It states the asymptotic behavior of the thermodynamic magnetization for sequences lying in the phasecoexistence region for all sufficiently large and converging either to a secondorder point or to the tricritical point. The asymptotic behavior is expressed in terms of the unique positive, global minimum point of an associated polynomial that is introduced in hypothesis (iii) of the theorem. With several modifications the hypotheses of the next theorem are also the hypotheses under which we derive the rates at which later in the paper.
As shown in part (iii) of Theorem 3.1, the asymptotics of depend on the asymptotics of the scaled freeenergy function . Because of Lemma 7.2, the asymptotics of the finitesize magnetization in Theorems 4.1–4.3 depend on precisely the same asymptotics. Lemma 7.2 coincides with Lemma 4.1 in [9]. In that paper the connections among the asymptotics of the scaled freeenergy functional, the limit theorems underlying the asymptotics of the finitesize magnetization and Lemma 4.1 are described in detail. These limit theorems are analogues of the MDP in Theorem 7.1 and of the weak convergence limit in Theorem 8.1.
Theorem 3.1 restates the main theorem in [13], Theorem 3.2. Hypotheses (iii)(a) and (iv) in the next theorem coincide with hypotheses (iii)(a) and (iv) in Theorem 3.2 in [13] except that the latter hypotheses are expressed in terms of and while here we have substituted the formulas for and . Hence and no longer appear.
Theorem 3.1
Let be a positive sequence that converges either to a secondorder point , , or to the tricritical point . We assume that satisfies the following four hypotheses: {longlist} lies in the phasecoexistence region for all sufficiently large . The sequence is parametrized by . This parameter regulates the speed of approach of to the secondorder point or the tricritical point in the following sense: both exist, and and are not both ; if , then equals or . There exists an even polynomial of degree or satisfying as together with the following two properties; is called the Ginzburg–Landau polynomial. {longlist} [(a) (iii)] (a) and such that for all uniformly for in compact subsets of . (b) has a unique, positive global minimum point ; thus the set of global minimum points of equals or . (iv) There exists a polynomial satisfying as together with the following property: such that sufficiently large and satisfying , . Under hypotheses (i)–(iv), for any If , then this becomes .
It is clear from the proof of the theorem that if hypotheses (iii) and (iv) are valid for a specific value of , then we obtain the asymptotic formula for that value of .
In the next section, we state the main results on the rates at which for small satisfying , for large satisfying , and for intermediate satisfying . We also compare these rates with the asymptotic behavior of the thermodynamic magnetization .
4 Main results on rates at which
Let be a positive sequence converging to 0. In stating the three results on the rates at which the finitesize magnetization , we write
and we write
Let be the quantity parametrizing the sequences as explained in hypothesis (ii) of Theorem 3.1. We begin with Theorem 4.1, which gives the rate at which for small satisfying . Theorem 4.2 gives the rate at which for large satisfying while Theorem 4.3 gives the rate at which for intermediate satisfying . In all three cases we compare these rates with the rate at which . In the next section we specialize these theorems to the six sequences mentioned in the Introduction.
Part (a) of the next theorem gives the rate at which for , and part (b) shows that for these values of , . It follows that for , is a physically relevant estimator of the finitesize magnetization because it has the same asymptotic behavior as that quantity.
The next theorem is valid under hypotheses (i) and (ii) of Theorem 3.1, hypotheses (iii)(a) and (iv) of that theorem for all , the inequality , and a new hypothesis (iii)(b). The inequality is satisfied by all six sequences considered in Section 5. The new hypothesis (iii)(b) restricts hypothesis (iii)(b) of Theorem 3.1 by assuming that the set of global minimum points of the Ginzburg–Landau polynomial equals for some . As we remark after the statement of the theorem, this restriction is needed in order to prove part (a). The proof does not cover the case where the set of global minimum points of equals for some . The conjecture is that in this case there exists such that (see the discussion before Corollary 7.3). An example of a sequence for which the set of global minimum points of contains three points is given in case (d) of sequence 4 in the next section. By contrast, all the other sequences considered in the next section satisfy the new hypothesis that the set of global minimum points of equals for some .
Theorem 4.1 (())
Let be a positive sequence parametrized by and converging either to a secondorder point , , or to the tricritical point . We assume hypotheses (i) and (ii) of Theorem 3.1 together with hypotheses (iii)(a) and (iv) of that theorem for all . We also assume the inequality and the following hypothesis, which restricts hypothesis (iii)(b) of Theorem 3.1: {longlist}[(iii)(b)] (iii)(b) The set of global minimum points of the Ginzburg–Landau polynomial equals for some . The following conclusions hold: (a) For all
(b) For all , .
Part (a) of the theorem is proved from the moderate deviation principle (MDP) for the distributions of in Theorem 7.1, which shows that the rate function equals . The inequality is used to control an error term in the proof of the MDP. According to hypothesis (iii)(b), the set of global minimum points of equals for some . It quickly follows from the MDP that the sequence of distributions of converges weakly to