Large deviation principle for empirical fields of log and Riesz gases.(English)Zbl 1397.82007
The present paper is devoted to the study a system of \(N\) particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. The authors are interested in investigating a microscopic quantity, the tagged empirical field, for which they establish a large deviation principle at speed \(N\). The rate function is found as well. Moreover, a variational property of the sine-beta processes is obtained which arise in random matrix theory. A next-to-leading order expansion of the free energy of the system is found which allows to show the existence of the thermodynamic limit.
Reviewer: Farruh Mukhamedov (Kuantan)
MSC:
| 82B05 | Classical equilibrium statistical mechanics (general) |
| 82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 15B52 | Random matrices (algebraic aspects) |
| 60F10 | Large deviations |
Cite
References:
| [1] | Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Wasserstein Space of Probability Measures. Birkäuser, Basel (2005) ·Zbl 1090.35002 |
| [2] | Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random normal matrices. Duke Math. J. 159(1), 31-81 (2011) ·Zbl 1225.15030 |
| [3] | Ameur, Yacin, Hedenmalm, Haakan, Makarov, Nikolai: Random normal matrices and Ward identities. Ann. Probab. 43(3), 1157-1201 (2015) ·Zbl 1388.60020 |
| [4] | Alastuey, A., Jancovici, B.: On the classical two-dimensional one-component Coulomb plasma. J. Phys. 42(1), 1-12 (1981) |
| [5] | Ameur, Y., Ortega-Cerdà, J.: Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates. J. Funct. Anal. 263, 1825-1861 (2012) ·Zbl 1256.31001 |
| [6] | Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108(4), 517-542 (1997) ·Zbl 0954.60029 |
| [7] | Ben Arous, G., Zeitouni, O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123-174 (1998) ·Zbl 0916.60022 |
| [8] | Barré, J., Bouchet, F., Dauxois, T., Ruffo, S.: Large deviation techniques applied to systems with long-range interactions. J. Stat. Phys. 119(3-4), 677-713 (2005) ·Zbl 1170.82302 |
| [9] | Bethuel, F., Brezis, H., Hélein, F., Vortices, G.-L.: Progress in Nonlinear Differential Equations and their Applications, vol. 13. Birkhäuser Boston Inc., Boston (1994) ·Zbl 0802.35142 |
| [10] | Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: Local density for two-dimensional one-component plasma (2015). arxiv:1510.02074 ·Zbl 1383.82056 |
| [11] | Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: The two-dimensional coulomb plasma: quasi-free approximation and central limit theorem (2016). arXiv:1609.08582 ·Zbl 1486.82042 |
| [12] | Bourgade, P., Erdös, L., Yau, H.T.: Bulk universality of general \[\beta\] β-ensembles with non-convex potential. J. Math. Phys. 53(9), 095221 (2012) ·Zbl 1278.82032 |
| [13] | Bourgade, P., Erdös, L., Yau, H.-T.: Universality of general \[\beta\] β-ensembles. Duke Math. J. 163(6), 1127-1190 (2014) ·Zbl 1298.15040 |
| [14] | Bekerman, F., Figalli, A., Guionnet, A.: Transport maps for Beta-matrix models and universality. arXiv preprint (2013). arXiv:1311.2315 ·Zbl 1330.49046 |
| [15] | Bodineau, T., Guionnet, A.: About the stationary states of vortex systems. Ann. Inst. H. Poincaré Probab. Stat. 35(2), 205-237 (1999) ·Zbl 0920.60095 |
| [16] | Borot, G., Guionnet, A.: All-order asymptotic expansion of beta matrix models in the multi-cut regime (2013). arXiv preprint arXiv:1303.1045 ·Zbl 1344.60012 |
| [17] | Borot, G., Guionnet, A.: Asymptotic expansion of \[\beta\] β matrix models in the one-cut regime. Commun. Math. Phys 317(2), 447-483 (2013) ·Zbl 1344.60012 |
| [18] | Borodachev, S., Hardin, D.H., Saff, E.B.: Minimal discrete energy on the sphere and other manifolds (2017) (in preparation) |
| [19] | Brauchart, J.S., Hardin, D.P., Saff, E.B.: The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere. In: Arvesú, J., López Lagomasino, G. (eds.) Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, vol. 578 of Contemporary Mathematics, pp. 31-61. American Mathematical Society, Providence (2012) ·Zbl 1318.31011 |
| [20] | Blanc, X., Lewin, M.: The crystallization conjecture: a review (2015). arXiv preprint arXiv:1504.01153 ·Zbl 1341.82010 |
| [21] | Bekerman, F., Leblé, T., Serfaty, S.: CLT for fluctuations of \[\beta\] β-ensembles with general potential (2017) (in preparation) ·Zbl 1406.60036 |
| [22] | Borodin, A., Sinclair, C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys 291(1), 177-224 (2009) ·Zbl 1184.82004 |
| [23] | Brush, S.G., Sahlin, H.L., Teller, E.: Monte-Carlo study of a one-component plasma. J. Chem. Phys 45, 2102-2118 (1966) |
| [24] | Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4-5), 383-402 (1998) ·Zbl 0928.49030 |
| [25] | Choquard, P., Clerouin, J.: Cooperative phenomena below melting of the one-component two-dimensional plasma. Phys. Rev. Lett. 50(26), 2086 (1983) |
| [26] | Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480(3), 57-159 (2009) |
| [27] | Chafaï, D., Gozlan, N., Zitt, P.-A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371-2413 (2014) ·Zbl 1304.82050 |
| [28] | Choquet, G.: Diamètre transfini et comparaison de diverses capacités. Technical report, Faculté des Sciences de Paris (1958) |
| [29] | Caillol, J.-M., Levesque, D., Weis, J.-J., Hansen, J.-P.: A monte carlo study of the classical two-dimensional one-component plasma. J. Stat.Phys. 28(2), 325-349 (1982) |
| [30] | Caffarelli, L.A., Riviere, N.M.: Smoothness and analyticity of free boundaries in variational inequalities. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3(2), 289-310 (1976) ·Zbl 0363.35009 |
| [31] | Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. PDE 32(7-9), 1245-1260 (2007) ·Zbl 1143.26002 |
| [32] | Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian. Invent. Math. 171(2), 425-461 (2008) ·Zbl 1148.35097 |
| [33] | Daley, D.J., Verey-Jones, D.: An Introduction to the Theory of Point Processes. Springer, New York (1988) ·Zbl 0657.60069 |
| [34] | Daley, D.J., Verey-Jones, D.: An Introduction to the Theory of Point Processes, vol. II. Springer, Berlin (2008) ·Zbl 1159.60003 |
| [35] | Dyson, F.: Statistical theory of the energy levels of a complex system, part i. J. Math. Phys. 3, 140-156 (1962) ·Zbl 0105.41604 |
| [36] | Dyson, F.: Statistical theory of the energy levels of a complex system, part ii. J. Math. Phys. 3, 157-185 (1962) ·Zbl 0105.41604 |
| [37] | Dyson, F.: Statistical theory of the energy levels of a complex system, part iii. J. Math. Phys. 3, 166-175 (1962) ·Zbl 0105.41604 |
| [38] | Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, vol. 38 of Stochastic Modelling and Applied Probability. Springer, Berlin, 2010. Corrected reprint of the second edition (1998) ·Zbl 0896.60013 |
| [39] | Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77-116 (1982) ·Zbl 0498.35042 |
| [40] | Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16(3), 961-977 (1988) ·Zbl 0648.60028 |
| [41] | Föllmer, H.: Random fields and diffusion processes. In: École d’Été de Probabilités de Saint-Flour XV-XVII, 1985-87, vol. 1362 of Lecture Notes in Mathematics, pp. 101-203. Springer, Berlin (1988) ·Zbl 0661.60063 |
| [42] | Forrester, P.J.: Exact integral formulas and asymptotics for the correlations in the \[1/r^21\]/r2 quantum many-body system. Phys. Lett. A 179(2), 127-130 (1993) |
| [43] | Forrester, P.J.: Log-Gases and Random Matrices, London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010) ·Zbl 1217.82003 |
| [44] | Frostman, O.: Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddelanden Mat. Sem. Univ. Lund 3, 115 s, (1935) ·JFM 61.1262.02 |
| [45] | Georgii, H.-O.: Large deviations and maximum entropy principle for interacting random fields on. Ann. Probab. 21(4), 1845-1875 (1993) ·Zbl 0790.60031 |
| [46] | Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440-449 (1965) ·Zbl 0127.39304 |
| [47] | Girvin, S.: Introduction to the fractional quantum Hall effect. In: Douçot, B., Pasquier, V., Duplantier, B., Rivasseau, V. (eds.) The Quantum Hall Effect, pp. 133-162. Springer, Berlin (2005) ·Zbl 1148.35097 |
| [48] | Georgii, H.-O., Zessin, H.: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96(2), 177-204 (1993) ·Zbl 0792.60024 |
| [49] | Hardin, D.P., Leblé, T., Saff, E.B., Serfaty, S.: Large deviation principles for hypersingular Riesz gases (2017). arXiv:1702.02894 ·Zbl 1398.82044 |
| [50] | Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, vol. 77. American Mathematical Society, Providence (2000) ·Zbl 0955.46037 |
| [51] | Jancovici, B., Lebowitz, J., Manificat, G.: Large charge fluctuations in classical Coulomb systems. J. Stat. Phys 72(3-4), 773-777 (1993) ·Zbl 1101.82307 |
| [52] | Kinderlehrer, D.: Variational inequalities and free boundary problems. Bull. Am. Math. Soc. 84(1), 7-26 (1978) ·Zbl 0382.35004 |
| [53] | Kapfer, S., Krauth, W.: Two-dimensional melting: from liquid-hexatic coexistence to continuous transitions. Phys. Rev. Lett. 114(3), 035702 (2015) |
| [54] | Killip, R., Stoiciu, M.: Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146(3), 361-399 (2009) ·Zbl 1155.81020 |
| [55] | Leblé, T.: Local microscopic behavior for 2D Coulomb gases. Probab. Theor. Relat. Fields (2016). doi:10.1007/s00440-016-0744-y ·Zbl 1379.82004 |
| [56] | Leblé, T.: Logarithmic, Coulomb and Riesz energy of point processes. J. Stat. Phys. 162(4), 887-923 (2016) ·Zbl 1341.82021 |
| [57] | Li, Y.: Rigidity of eigenvalues for beta-ensemble in multi-cut regime (2016). arXiv preprint arXiv:1611.06603 |
| [58] | Lieb, E.H., Lebowitz, J.L.: Existence of thermodynamics for real matter with Coulomb forces. Phys. Rev. Lett. 22, 631-634 (1969) |
| [59] | Lieb, E.H., Narnhofer, H.: The thermodynamic limit for jellium. J. Stat. Phys. 12, 291-310 (1975) ·Zbl 0973.82500 |
| [60] | Lieb, E., Rougerie, N., Yngvason, J.: Local incompressibility estimates for the Laughlin phase (2017). arXiv preprint arXiv:1701.09064 ·Zbl 1410.82009 |
| [61] | Leblé, T., Serfaty, S.: Fluctuations of two-dimensional Coulomb gases (2016). arXiv preprint arXiv:1609.08088 ·Zbl 1423.60045 |
| [62] | Leblé, T., Serfaty, S., Zeitouni, O.: Large deviations for the two-dimensional two-component plasma. Commun. Math. Phys. 350(1), 301-360 (2017) ·Zbl 1371.82109 |
| [63] | Mazars, M.: Long ranged interactions in computer simulations and for quasi-2d systems. Phys. Rep. 500, 43-116 (2011) |
| [64] | Nakano, F.: Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble. J. Stat. Phys. 156(1), 66-93 (2014) ·Zbl 1303.82019 |
| [65] | Petrache, M., Rota-Nodari, S.: Equidistribution of jellium energy for Coulomb and Riesz interactions (2016). arXiv preprint arXiv:1609.03849 ·Zbl 1391.82003 |
| [66] | Penrose, O., Smith, E.R.: Thermodynamic limit for classical systems with Coulomb interactions in a constant external field. Commun. Math. Phys 26, 53-77 (1972) |
| [67] | Petrache, M., Serfaty, S.: Next order asymptotics and renormalized energy for Riesz interactions. J. Inst. Math. Jussieu. 16(3), 501-569 (2017) ·Zbl 1373.82013 |
| [68] | Rota Nodari, S., Serfaty, S.: Renormalized energy equidistribution and local charge balance in 2D Coulomb systems. Int. Math. Res. Not. 2015(11), 3035-3093 (2015) ·Zbl 1321.82029 |
| [69] | Rassoul-Agha, F., Seppäläinen, T., A course on Large Deviation Theory with an Introduction to Gibbs Measures, volume 162 of Graduate Studies in Mathematics, 2015 edn. American Mathematical Society (2009) ·Zbl 1330.60001 |
| [70] | Rougerie, N., Serfaty, S.: Higher-dimensional Coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. 69(3), 519-605 (2016) ·Zbl 1338.82043 |
| [71] | Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. 2007 (2007). doi:10.1093/imrn/rnm006 ·Zbl 1130.60030 |
| [72] | Serfaty, S.: Coulomb gases and Ginzburg-Landau vortices. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2015) ·Zbl 1335.82002 |
| [73] | Shcherbina, M.: Fluctuations of linear eigenvalue statistics of \[\beta\] β matrix models in the multi-cut regime. J. Stat. Phys 151(6), 1004-1034 (2013) ·Zbl 1273.15042 |
| [74] | Saff, E., Kuijlaars, A.: Distributing many points on a sphere. Math. Intell. 19(1), 5-11 (1997) ·Zbl 0901.11028 |
| [75] | Sari, R., Merlini, D.: On the \[\nu\] ν-dimensional one-component classical plasma: the thermodynamic limit problem revisited. J. Stat. Phys. 14(2), 91-100 (1976) |
| [76] | Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model. In: Progress in Nonlinear Differential Equations and their Applications, vol. 70, Birkhäuser, Boston (2007) ·Zbl 1112.35002 |
| [77] | Sandier, E., Serfaty, S.: From the Ginzburg-Landau model to vortex lattice problems. Commun. Math. Phys. 313, 635-743 (2012) ·Zbl 1252.35034 |
| [78] | Sandier, E., Serfaty, S.: 1d log gases and the renormalized energy: crystallization at vanishing temperature. Probab. Theory Relat. Fields 162(3-4), 795-846 (2015) ·Zbl 1327.82005 |
| [79] | Sandier, E., Serfaty, S.: 2d Coulomb gases and the renormalized energy. Ann. Probab. 43(4), 2026-2083 (2015) ·Zbl 1328.82006 |
| [80] | Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenchaften vol. 316, Springer, Berlin (1997) ·Zbl 0881.31001 |
| [81] | Stormer, H., Tsui, D., Gossard, A.: The fractional quantum Hall effect. Rev. Mod. Phys. 71(2), S298 (1999) |
| [82] | Stishov, S.M.: Does the phase transition exist in the one-component plasma model? J. Exp. Theor. Phys. Lett. 67(1), 90-94 (1998) |
| [83] | Torquato, S.: Hyperuniformity and its generalizations. Phys. Rev. E 94(2), 022122 (2016) |
| [84] | Varadhan, S.R.S.: Large deviations and applications. In: École d’Été de Probabilités de Saint-Flour XV-XVII, 1985-87, vol. 1362 of Lecture Notes in Mathematics, pp. 1-49. Springer, Berlin (1988) ·Zbl 0661.60040 |
| [85] | Valkó, B., Virág, B.: Continuum limits of random matrices and the brownian carousel. Invent. Math. 177(3), 463-508 (2009) ·Zbl 1204.60012 |
| [86] | Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548-564 (1955) ·Zbl 0067.08403 |
| [87] | Zabrodin, A., Wiegmann, P.: Large-\[NN\] expansion for the 2D Dyson gas. J. Phys. A 39(28), 8933-8963 (2006) ·Zbl 1098.82011 |
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