Electronic Journal of Probability

A tightness criterion for random fields, with application to the Ising model

Marco Furlan and Jean-Christophe Mourrat

Full-text: Open access

Abstract

We present a criterion for a family of random distributions to be tight in local Hölder and Besov spaces of possibly negative regularity on general domains. We then apply this criterion to find the sharp regularity of the magnetization field of the two-dimensional Ising model at criticality, answering a question of [8].

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 97, 29 pp.

Dates
Received: 2 November 2016
Accepted: 25 October 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1510802251

Digital Object Identifier
doi:10.1214/17-EJP121

Mathematical Reviews number (MathSciNet)
MR3724565

Zentralblatt MATH identifier
1378.60065

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G60: Random fields 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
tightness criterion local Besov spaces Ising model

Rights
Creative Commons Attribution 4.0 International License.

Citation

Furlan, Marco; Mourrat, Jean-Christophe. A tightness criterion for random fields, with application to the Ising model. Electron. J. Probab. 22 (2017), paper no. 97, 29 pp. doi:10.1214/17-EJP121. https://projecteuclid.org/euclid.ejp/1510802251


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References

  • [1] Abdesselam, A.: A Second-Quantized Kolmogorov-Chentsov Theorem, arXiv:1604.05259
  • [2] Armstrong, S. and Kuusi, T. and Mourrat, J.-C.: The additive structure of elliptic homogenization. Invent. Math. 208 (3), (2017), 999–1154
  • [3] Armstrong, S. and Kuusi, T. and Mourrat, J.-C.: Quantitative stochastic homogenization and large-scale regularity. Preliminary version available at http://perso.ens-lyon.fr/jean-christophe.mourrat/lecturenotes.pdf.
  • [4] H. Bahouri, H. and Chemin, J.-Y. and Danchin. R.: Fourier analysis and nonlinear partial differential equations. Springer, Heidelberg, 2011. xvi+523 pp.
  • [5] Bauerschmidt, R. and Bourgade, P. and Nikula, M and Yau, H.-T.: The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem, arXiv:1609.08582
  • [6] J. Bergh, J. and Löfström, J.: Interpolation spaces: an introduction, Springer-Verlag, Berlin, Heidelberg, 1976. x+207 pp.
  • [7] Biskup, M. and Spohn, H.: Scaling limit for a class of gradient fields with nonconvex potentials. Ann. Probab. 39 (1), (2011), 224–251
  • [8] Camia, F. and Garban, C. and Newman, C.M.: Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43 (2), (2015), 528–571
  • [9] Camia, F. and Garban, C. and Newman, C.M.: Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits. Ann. Inst. H. Poincaré Probab. Statist. 52 (1), (2016), 146–161
  • [10] Duminil-Copin, H. and Hongler, C. and Nolin, P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 (9), (2011), 1165–1198
  • [11] Gerencsér, M. and Hairer, M.: Singular SPDEs in domains with boundaries. arXiv:1702:06522
  • [12] Chelkak, D. and Hongler, C. and Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181 (3), (2015), 1087–1138
  • [13] Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (7), (1988), 909–996
  • [14] Edwards, R.G. and Sokal, A.D.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38 (6), (1988), 2009–2012
  • [15] Fortuin, C.M. and Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, (1972), 536–564
  • [16] Gu, Y. and Mourrat, J.-C.: Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul. 14 (1), (2016), 452–481
  • [17] Giacomin, G. and Olla, S. and Spohn, H.: Equilibrium fluctuations for $\nabla \varphi $ interface model. Ann. Probab. 29 (3), (2001), 1138–1172
  • [18] Grimmett, G.R.: The Random-Cluster Model. Springer-Verlag, Berlin, 2009.
  • [19] Hairer, M.: A theory of regularity structures. Invent. Math. 198 (2), (2014), 269–504
  • [20] Hairer, M. and Labbé, C.: Multiplicative stochastic heat equations on the whole space. Journal of the Europ. Math. Soc. arXiv:1504.07162
  • [21] Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29 (3), (2001), 1128–1137
  • [22] Leblé, T. and Serfaty., S.: Fluctuations of two-dimensional Coulomb gases. arXiv:1609.08088
  • [23] Meyer, Y.: Wavelets and operators. Transl. D.H. Salinger. Cambridge University Press, (1992). xvi+224 pp.
  • [24] Mourrat, J-C. and Nolen, J.: Scaling limit of the corrector in stochastic homogenization. Ann. Appl. Probab. 27 (2), (2017), 944–959
  • [25] Mourrat, J-C. and Otto, F.: Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44 (5), (2016), 3207–3233
  • [26] Mourrat, J-C. and Weber, H.: Convergence of the two-dimensional dynamic Ising–Kac model to $\Phi ^4_2$. Comm. Pure Appl. Math. 70, (2017), 717–812
  • [27] Naddaf, A. and Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 (1), (1997), 55–84
  • [28] Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65 (2), (1944), 117–149
  • [29] Pinsky, M.A.: Introduction to Fourier analysis and wavelets. American Mathematical Society, Providence, RI, 2009. xx+376 pp.
  • [30] Triebel, H.: Theory of function spaces III Birkhäuser Verlag, Basel, 2006. xii+426 pp.