Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Path-space moderate deviations for a Curie–Weiss model of self-organized criticality

Francesca Collet, Matthias Gorny, and Richard C. Kraaij

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The dynamical Curie–Weiss model of self-organized criticality (SOC) was introduced in (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678) and it is derived from the classical generalized Curie–Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie–Weiss model of SOC (Ann. Probab. 44 (2016) 444–478) as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton–Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.


Le modèle de Curie–Weiss de criticalité auto-organisée dynamique a été construit dans (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678) à partir du modèle de Curie–Weiss généralisé. Il s’agit d’un processus de Markov continu dont l’unique mesure invariante est la loi du modèle de Curie–Weiss de criticalité auto-organisée (Ann. Probab. 44 (2016) 444–478). Dans le cas Gaussien, nous étudions les fluctuations modérées de la magnétisation. Nous obtenons un principe de déviations modérées dans l’espace des chemins en utilisant une approche analytique basée sur la convergence de générateurs non-linéaires et sur l’unicité des solutions de viscosité pour des équations de Hamilton–Jacobi associées. Notre résultat montre que, dans une certaine échelle de temps modérée et sans intervention de paramètres extérieurs, le comportement critique de la magnétisation est critique.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 765-781.

Received: 26 January 2018
Revised: 23 November 2018
Accepted: 20 March 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60F10: Large deviations 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Moderate deviations Interacting particle systems Mean-field interaction Self-organized criticality Hamilton–Jacobi equation Perturbation theory for Markov processes


Collet, Francesca; Gorny, Matthias; Kraaij, Richard C. Path-space moderate deviations for a Curie–Weiss model of self-organized criticality. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 765--781. doi:10.1214/19-AIHP981.

Export citation


  • [1] M. J. Aschwanden (Ed.). Self-Organized Criticality Systems. Open Academic Press, Berlin, 2013.
  • [2] P. Bak. Complexity and criticality. In How Nature Works: The Science of Self-Organized Criticality 1–32. Springer Science $+$ Business Media, New York, 1996.
  • [3] P. Bak and K. Sneppen. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett. 71 (1993) 4083–4086.
  • [4] P. Bak, C. Tang and K. Wiesenfeld. Self-organized criticality: An explanation of the $1/f$ noise. Phys. Rev. Lett. 59 (1987) 381–384.
  • [5] R. Cerf and M. Gorny. A Curie–Weiss model of self-organized criticality. Ann. Probab. 44 (2016) 444–478.
  • [6] F. Collet and R. C. Kraaij. Dynamical moderate deviations for the Curie–Weiss model. Stochastic Process. Appl. 127 (2017) 2900–2925.
  • [7] F. Collet and R. C. Kraaij. Path-space moderate deviation principles for the random field Curie–Weiss model. Electron. J. Probab. 23 (2018) 1–45 (paper no. 21).
  • [8] X. Deng, J. Feng and Y. Liu. A singular 1-D Hamilton–Jacobi equation, with application to large deviation of diffusions. Commun. Math. Sci. 9 (2011) 289–300.
  • [9] D. Dhar. Theoretical studies of self-organized criticality. Phys. A 369 (2006) 29–70.
  • [10] R. S. Ellis and C. M. Newman. Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 (1978) 117–139.
  • [11] W. Feller. An Introduction to Probability Theory and Its Applications. Volume I, 3rd edition. John Wiley & Sons Inc., New York, 1968.
  • [12] J. Feng, J.-P. Fouque and R. Kumar. Small-time asymptotics for fast mean-reverting stochastic volatility models. Ann. Appl. Probab. 22 (2012) 1541–1575.
  • [13] J. Feng and T. G. Kurtz. Large Deviations for Stochastic Processes. Mathematical Surveys and Monographs 131. American Mathematical Society, Providence, RI, 2006.
  • [14] M. Gorny. A Curie–Weiss model of self-organized criticality: The Gaussian case. Markov Process. Related Fields 20 (2014) 563–576.
  • [15] M. Gorny. A dynamical Curie–Weiss model of SOC: The Gaussian case. Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 658–678.
  • [16] R. C. Kraaij. Large deviations for finite state Markov jump processes with mean-field interaction via the comparison principle for an associated Hamilton–Jacobi equation. J. Stat. Phys. 164 (2016) 321–345.
  • [17] T. G. Kurtz. Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics. Trans. Amer. Math. Soc. 186 (1973) 259–272.
  • [18] T. G. Kurtz. A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12 (1973) 55–67.
  • [19] R. Meester and A. Sarkar. Rigorous self-organised criticality in the modified Bak–Sneppen model. J. Stat. Phys. 149 (2012) 964–968.
  • [20] G. C. Papanicolaou, D. Stroock and S. R. S. Varadhan. Martingale approach to some limit theorems. In Statistical Mechanics, Dynamical Systems and the Duke Turbulence Conference. Volume 3. Duke University Series, Durham, NC, 1977.
  • [21] G. Pruessner. Self-Organised Criticality: Theory, Models and Characterisation. Cambridge University Press, Cambridge, 2012.
  • [22] B. Ráth and B. Tóth. Erdős–Rényi random graphs $+$ forest fires $=$ self-organized criticality. Electron. J. Probab. 14 (2009) 1290–1327 (paper no. 45).
  • [23] D. Sornette. Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools. Springer-Verlag, Berlin, 2006.
  • [24] D. L. Turcotte. Self-organized criticality. Rep. Progr. Phys. 62 (1999) 1377–1429.