Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models

This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, \textbf{165} (2016), no. 6, 1114--1140]. The proof is based on Stein's method for exchangeable pairs.


Introduction and main result
Let N ≥ 2 be an integer, and let S N −1 denote the unit sphere in R N . In this paper, we consider the mean-field classical N -vector spin models, where each spin σ i is in S N −1 , at a complete graph vertex i among n vertices ( [5,Chapter 9]). The state space is Ω n = (S N −1 ) n with product measure P n = µ × · · · × µ, where µ is the uniform probability measure on S N −1 . In the absence of an external field, each spin configuration σ = (σ 1 , . . . , σ n ) in the state space Ω n has a Hamiltonian defined by In the Heisenberg model (N = 3), Kirkpatrick and Meckes [6] established large deviation, normal approximation results for total spin S n = n i=1 σ i in the non-critical phase (β = 3), and a non-normal approximation result in the critical phase (β = 3). The results in [6] are generalized by Kirkpatrick and Nawaz [7] to the mean field N -vector models with N ≥ 2.
Let I ν denote the modified Bessel function of the first kind (see, e.g., [2, p. 713]) and f (x) = , x > 0.   Based on their large deviations, Kirkpatrick and Nawaz [7] argued that in the case β > N , there exists ε > 0 such that σ i is total spin. It means that |S n | is close to bn/β with high probability. On the other hand, all points on the hypersphere of radius bn/β will have equal probability due to symmetry. Based on these facts, they considered the fluctuations of the squared-length of total spin: (1.6) Kirkpatrick and Nawaz [7] proved that when β > N , the bounded-Lipschitz distance between W n /B and Z is bounded by C(log n/n) 1/4 . Their proof is based on Stein's method for exchangeable pairs (see Stein [10]). Recall that a random vector (W, W ) is called an exchangeable pair if (W, W ) and (W , W ) have the same distribution. Kirkpatrick and Nawaz [7] construct an exchangeable pair as follows. Let W n be as in (1.5) and let σ = {σ 1 , . . . , σ n }, where for each i fixed, σ i is an independent copy of σ i given {σ j , j = i}, i.e., given {σ j , j = i}, σ i and σ i have the same distribution and σ i is conditionally independent of σ i (see, e.g., [4, p. 964]). Let I be a random index independent of all others and uniformly distributed over {1, . . . , n}, and let The bound C(log n/n) 1/4 obtained by Kirkpatrick and Nawaz [7] is not sharp. The aim of this paper is to give the Kolmogorov and Wasserstein distances between W n /B and Z with optimal rate Cn −1/2 .
The main result is the following theorem. We recall that, throughout this paper, C is a positive constant which depends only on β, and its value may be different for each appearance.   The Wasserstein bound in Theorem 1.1 will be a consequence of the following proposition, a version of Stein's method for exchangeable pairs. It is a special case of Theorem 2.4 of Eichelsbacher and Löwe [4] or Theorem 13.1 in [3]. Proposition 1.2. Let (W, W ) be an exchangeable pair and ∆ = W − W . If E(∆|W ) = λ(W + R) for some random variable R and 0 < λ < 1, then The Kolmogorov distance is more commonly used in probability and statistics, and is usually more difficult to handle than the Wasserstein distance. Recently, Shao and Zhang [9] proved a very general theorem. Their result is as follows. Proposition 1.3. Let (W, W ) be an exchangeable pair and ∆ = W − W . Let ∆ * := ∆ * (W, W ) be any random variable satisfying ∆ * (W, W ) = ∆ * (W , W ) and ∆ * ≥ |∆|. If E(∆|W ) = λ(W + R) for some random variable R and 0 < λ < 1, then Shao and Zhang [9] applied their bound in Proposition 1.3 to get optimal bound in many problems, including a bound of O(n −1/2 ) for the Kolmogorov distance in normal approximation of total spin in the Heisenberg model. We note that if |∆| ≤ a, then the following result is an immediate corollary of Proposition 1.3. In this case, the bound is much simpler than that of Proposition 1.3.
For S n = n i=1 σ i , and for W n and W n respectively defined in (1.5) and (1.7), we have since |S n |+|S n | ≤ 2n and |S n |−|S n | ≤ |σ I −σ I | ≤ 2. Therefore, we will apply Corollary 1.4 to obtain the Kolmogorov bound in Theorem 1.1.

Proof of the main result
The proof of Theorem 1.1 depends on Kirkpatrick and Nawaz's finding [7]. Applying Proposition 1.2 and Corollary 1.4, Theorem 1.1 follows from the following proposition. Then the following statements hold:

Remark 2.2.
Kirkpatrick and Nawaz's [7] used their large deviation result for total spin S n to prove that EW 2 n ≤ C log n. Intuitively, we see that this bound would be improved to EW 2 n ≤ C since W n approximates a normal distribution. By a more careful estimate, we can prove that E (β|S n |/n − b) 2 ≤ C/n (see Lemma A.1). This will lead to desired bound EW 2 n ≤ C. Kirkpatrick and Nawaz's [7] also proved that To get optimal bound of order n −1/2 for this term, we use a fine estimate of function The proof of the first half of (i) is completed. Now, apply Lemma A.1 given in the Appendix, we have (ii) Kirkpatrick and Nawaz [7, equation (9)] showed that where R 1 is a random variable satisying E|R 1 | ≤ Cn −3/2 . Set g(x) = xf (x), x > 0. By Taylor's expansion, we have for some positive random variable ξ: By Lemma A.2 (ii), we have |g (ξ)| < 6. Since V ≥ 1, EW 2 n ≤ C and E|R 1 | ≤ Cn −3/2 , we conclude that E|R| ≤ Cn −1/2 . The proof of (ii) is completed.
(iii) Denote Id is the n × n identity matrix and set . From Kirkpatrick and Nawaz [7,Equations (11) and (12)], we have and P i is orthogonal projection onto r i . Therefore, Thus, Normal approximation for the mean field classical N -vector models To bound E|R 5 |, we note that E|b i − b| (by Lemma A.2 (iii) and the fact that |σ (i) | ≤ n) (2.5) and the fact that ||σ (i) | − |S n || ≤ 1). (2.8) Similarly, It follows that Define a probability density function where Z 2 12 is the normalizing constant. Let (ξ 1 , ξ 2 ) ∼ p 12 (x, y) given (σ j ) j>2 , and for i = 1, 2Ṽ and Using similar estimate for Ṽ 2 − a , then we have ≤ Cn 3 (by Lemma A.1). (2.17) Normal approximation for the mean field classical N -vector models Note that given (σ j ) j>2 , ξ 1 and ξ 2 are conditionally independent. It implies that The proposition is proved.

A Appendix
In this Section, we will prove the technical results that used in the proof of Theorem 1.1.
Proof. By the large deviation for S n /n [7,Proposition 2] and the argument in [7, p. 1126], one can prove that there exists ε > 0 such that . Then the following statements hold: Proof. As was showed in [7, p. 1134], we have Amos [1, p. 243] proved that x ≤ 4 + 2f 2 (x) (by the first half of (A.5)) ≤ 6 (by the second half of (A.5)).
The proof of (i) and (ii) is completed. For (iii), we have x .
Apply Theorem 2 (a) of Nȧsell [8], we can show that The proof of (iii) is completed.
Lemma A.3. With the notation in the proof of Theorem 1.1, we havẽ Proof. Let A N = 2π N/2 /Γ(N/2) the Lebesgue measure of S N −1 . It follows from (2.12) where we have used formula exp(z cos θ) sin 2ν θdθ (see, e.g., Exercise 11.5.4 in [2]) in the last equation. For i = 1, 2, we havẽ Finally, we would like to note again that Proposition 1.2 is a special case of Theorem 2.4 of Eichelsbacher and Löwe [4] or Theorem 13.1 in [3], but the constants in the bound may be different from those of Theorem 2.4 in [4] or Theorem 13.1 in [3]. Since the proof is short and simple, we will present here. 2λ By Lemma 2.4 in [3] we have f ≤ 2, f ≤ 2/π, f ≤ 2. (A.10) The conclusion of the proposition follows from (A.9) and (A.10).