Skip to main content

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

Abstract

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47–97 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Dembo, A., Montanari, A.: Ising models on locally tree-like graphs. Ann. Appl. Probab. 20(2), 565–592 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Dembo, A., Montanari, A., Sly, A., Sun, N.: The replica symmetric solution for Potts models on \(d\)-regular graphs. Commun. Math. Phys. 327, 551–575 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dommers, S., Giardinà, C., van der Hofstad, R.: Ising critical exponents on random trees and graphs. Commun. Math. Phys. 328(1), 355–395 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Dommers, S., Giardinà, C., Gilberti, C., van der Hofstad, R., Prioriello, M.L.: Ising critical behavior of imhomogeneous Curie–Weiss models and annealed random graphs. Commun. Math. Phys. 38, 221–263 (2016)

    ADS  Article  MATH  Google Scholar 

  6. 6.

    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E 66, 016104 (2002)

    ADS  Article  Google Scholar 

  7. 7.

    Camia, F., Newman, C.M., Garban, C.: The Ising magnetization exponent is \(\frac{1}{15}\). Probab. Theory Relat. Fields 160, 175–187 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Camia, F., Newman, C.M., Garban, C.: Planar Ising magnetization field I. Uniqueness of the scaling limit. Ann. Probab. 43, 528–571 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Can, V.H.: Annealed limit theorems for the Ising model on random regular graphs. arXiv:1701.08639 (2017)

  10. 10.

    Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften. Springer, New York (1985)

    Book  MATH  Google Scholar 

  11. 11.

    Ellis, R., Newman, C.: The statistics of Curie–Weiss models. J. Stat. Phys. 19, 149–161 (1978)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Ellis, R., Newman, C.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete. 44, 117–139 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Grimmett, G.: The Random-Cluster Model. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  14. 14.

    Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.L.: Quenched central limit theorems for the Ising model on random graphs. J. Stat. Phys. 160, 1623–1657 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Giardinà, C., Giberti, C., van der Hofstad, R., Prioriello, M.L.: Annealed central limit theorems for the Ising model on random graphs. ALEA Lat. Am. J. Probab. Math. Stat. 13(1), 121–161 (2016)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Mossel, E., Sly, A.: Exact thresholds for Ising–Gibbs samplers on general graphs. Ann. Probab. 41(1), 294–328 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    van der Hofstad, R.: Random graphs and complex networks. http://www.win.tue.nl/~rhofstad/NotesRGCN.html

Download references

Acknowledgements

We would like to thank the anonymous referees for their carefully reading and their valuable comments. This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.03–2017.01.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Van Hao Can.

Appendix: Proof of (54) and (55)

Appendix: Proof of (54) and (55)

We will repeat some computations in [9] and use Lemma 2.1 to prove these claims.

Proof of (54)

Using Stirling’s formula, we have

$$\begin{aligned} \left( {\begin{array}{c}n\\ j\end{array}}\right) = \left( \frac{1}{\sqrt{2 \pi }} + o(1) \right) \sqrt{\frac{n}{j(n-j)}} \exp \left( n I \left( \frac{j}{n}\right) \right) , \end{aligned}$$

with

$$\begin{aligned} I(t)=(t-1) \log (1-t) -t \log t. \end{aligned}$$

Therefore using (53), we get

$$\begin{aligned} \frac{x_j(n)}{x_{j_*}(n)}= & {} (1+o(1)) \sqrt{\frac{j_*(n-j_*)}{j(n-j)} } \exp \Bigg ( n \left[ I \left( \frac{j}{n}\right) - I \left( \frac{j_*}{n}\right) \right] \nonumber \\&+\, \log g(dj,dn) - \log g(dj_*,dn) \Bigg ) \nonumber \\= & {} (1+o(1))\sqrt{\frac{j_*(n-j_*)}{j(n-j)} } \exp \left( n \left[ I \left( \frac{j}{n}\right) - dF \left( \frac{j}{n}\right) \right] \right. \nonumber \\&\left. -\, n \left[ I \left( \frac{j_*}{n}\right) - d F \left( \frac{j_*}{n}\right) \right] + \left[ \log g(dj,dn)-ndF \left( \frac{j}{n}\right) \right] \right. \nonumber \\&\left. -\, \left[ \log g(dj_*,dn)-ndF \left( \frac{j_*}{n}\right) \right] \right) \nonumber \\= & {} (1+o(1))\sqrt{\frac{j_*(n-j_*)}{j(n-j)} } \exp \left( n \left[ H \left( \frac{j}{n}\right) - H \left( \frac{j_*}{n}\right) \right] \right. \nonumber \\&\left. + \left[ \log g(dj,dn)-ndF \left( \frac{j}{n}\right) \right] - \left[ \log g(dj_*,dn)-ndF \left( \frac{j_*}{n}\right) \right] \right) , \end{aligned}$$

which yields (54).

Proof of (55)

Since H(t) attains the maximum at a unique point \(\tfrac{1}{2}\), there exists a positive constant \(\varepsilon \), such that for all \(\delta \le \varepsilon \),

$$\begin{aligned} \max _{|t-\tfrac{1}{2}|\ge \delta } H(t) = \max \{H(\tfrac{1}{2} \pm \delta )\}. \end{aligned}$$

Hence for n large enough (such that \(n^{-1/6} \le \varepsilon \)), we have for all \(|j-j_*| > n^{5/6}\),

$$\begin{aligned} H(j/n)-H(j_*/n) \le \max \{H(j_*/n \pm n^{-1/6})- H(j_*/n) \}. \end{aligned}$$
(68)

Using the same arguments for (58), we can prove that

$$\begin{aligned} H\big (j_*/n \pm n^{-1/6}\big )- H(j_*/n) = \alpha _* n^{-2/3} + o\big (n^{-2/3}\big ). \end{aligned}$$

Therefore

$$\begin{aligned} n \Big (H\big (j_*/n \pm n^{-1/6}\big )- H(j_*/n)\Big ) = \alpha _* n^{1/3} +o\big (n^{1/3}\big ). \end{aligned}$$
(69)

Using \(\alpha _*\) and (68), (69), we have for n large enough and \(|j-j_*| > n^{5/6}\),

$$\begin{aligned} n \big (H(j/n)-H(j_*/n)\big ) \le \frac{\alpha _* n^{1/3}}{2}. \end{aligned}$$
(70)

On the other hand, for all j

$$\begin{aligned} \sqrt{\frac{j_*(n-j_*)}{j(n-j)}} \le \sqrt{n}. \end{aligned}$$
(71)

It follows from (54), (57), (70) and (71) that for n large enough and \(|j-j_*| > n^{5/6}\),

$$\begin{aligned} x_j (n) \le x_{j_*} (n) \sqrt{n} \exp \left( \frac{\alpha _* n^{1/3}}{2} \right) \le x_{j_*} (n) n^{-6}, \end{aligned}$$

since \(\alpha _*<0\). Therefore

$$\begin{aligned} \bar{A}_n:=\sum _{|j-j_*| > n^{5/6}} x_j(n) \le x_{j_*}(n) n^{-5}, \end{aligned}$$
(72)

here we recall that \(x_j(n)=0\) for all \(j<0\) or \(j >n\). Similarly, for n large enough and \(|j-j_*|> n^{5/6}\),

$$\begin{aligned} \exp \left( \frac{r(2j-n)}{n^{3/4}}\right) x_j(n)\le & {} x_{j_*}(n) \sqrt{n} \exp \left( \frac{|r(2j-n)|}{n^{3/4}}\right) \exp \left( \frac{\alpha _* n^{1/3}}{2}\right) \\\le & {} x_{j_*}(n) \sqrt{n} \exp \left( |r| n^{1/4}+ \frac{\alpha _* n^{1/3}}{2}\right) \\\le & {} x_{j_*}(n) n^{-6}. \end{aligned}$$

Hence

$$\begin{aligned} \bar{B}_n:= \sum _{|j-j_*| > n^{5/6}} \exp \left( \frac{r(2j-n)}{n^{3/4}}\right) x_j(n) \le x_{j_*}(n) n^{-5}. \end{aligned}$$
(73)

Since all the terms \((x_j(n))\) are non negative,

$$\begin{aligned} \hat{A}_n=\sum _{|j-j_*| \le n^{5/6}} x_j (n) \ge x_{j_*} (n), \end{aligned}$$
(74)

and

$$\begin{aligned} \hat{B}_n = \sum _{|j-j_*| \le n^{5/6}} \exp \left( \frac{r(2j-n)}{n^{3/4}}\right) x_j (n)\ge & {} \exp \left( \frac{r(2j_*-n)}{n^{3/4}}\right) x_{j_*} (n) \nonumber \\\ge & {} \exp \left( -\frac{|r|}{n^{3/4}} \right) x_{j_*}(n) \ge \frac{x_{j_*}(n)}{2}, \end{aligned}$$
(75)

for n large enough and r fixed. Finally, combining (72), (73), (74) and (75) yields that

$$\begin{aligned} \frac{A_n}{B_n} = \frac{\hat{A}_n + \bar{A}_n}{\hat{B}_n + \bar{B}_n} = \frac{\hat{A}_n}{\hat{B}_n } + o\big (n^{-2}\big ). \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Can, V.H. Critical Behavior of the Annealed Ising Model on Random Regular Graphs. J Stat Phys 169, 480–503 (2017). https://doi.org/10.1007/s10955-017-1879-7

Download citation

Keywords

  • Ising model
  • Random graphs
  • Critical behavior
  • Annealed measure

Mathematics Subject Classification

  • 05C80
  • 60F5
  • 82B20