zbMATH — the first resource for mathematics

1D log gases and the renormalized energy: crystallization at vanishing temperature. (English) Zbl 1327.82005
The statistical mechanics of a one-dimensional log gas or beta-ensemble with general potential and arbitrary beta (the inverse of temperature) is studied according to the method introduced by the same authors [Ann. Probab. 43, No. 4, 2026–2083 (2015; Zbl 1328.82006)] for 2D Coulomb gases. The formal limit beta going to infinity corresponds to the “weighted Fekete sets” and it is also treated her. A one-dimensional version of the renormalized energy of the same authors [Commun. Math. Phys. 313, No. 3, 635–743 (2012; Zbl 1252.35034)] is introduced that measures the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. It is shown that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian the full statistical mechanics problem is connected to this renormalized energy and this allows to obtain new results on the distribution of the points at the microscopic scale. In particular, it is shown that configurations whose \(W\) is above a certain threshold (tending to \(\min W\) as \(\beta\) goes to infinity) have exponentially small probability. This shows that such configurations have increasing when the temperature goes to zero.

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)
Full Text: DOI arXiv
[1] Akemann, G., Baik, J.P.: The Oxford Handbook of Random Matrix Theory. Oxford University Press, Di Francesco (2011) · Zbl 1225.15004
[2] Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010) · Zbl 1184.15023
[3] Alberti, G; Müller, S, A new approach to variational problems with multiple scales, Commun. Pure Appl. Math., 54, 761-825, (2001) · Zbl 1021.49012
[4] Aizenman, M; Martin, P, Structure of Gibbs states of one dimensional Coulomb systems, Commun. Math. Phys., 78, 99-116, (1980)
[5] Avila, A; Last, Y; Simon, B, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum, Anal. PDE, 3, 81-108, (2010) · Zbl 1225.26031
[6] Becker, ME, Multiparameter groups of measure-preserving transformations: a simple proof of wiener’s ergodic theorem, Ann. Probab., 9, 504-509, (1981) · Zbl 0468.28020
[7] Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Related Fields 108(4), 517-542 (1997) · Zbl 0954.60029
[8] Bekerman, F., Figalli, A., Guionnet, A.: Transport maps for \(β \)-matrix models and universality. arXiv:1311.2315 (2013) · Zbl 1330.49046
[9] Borodin, A; Serfaty, S, Renormalized energy concentration in random matrices, Commun. Math. Phys., 320, 199-244, (2013) · Zbl 1276.60007
[10] Borot, G; Guionnet, A, Asymptotic expansion of \(β \) matrix models in the one-cut regime, Commun. Math. Phys., 317, 447-483, (2013) · Zbl 1344.60012
[11] Borot, G; Guionnet, A, Asymptotic expansion of \(β \) matrix models in the multi-cut regime, Commun. Math. Phys., 317, 447-483, (2013) · Zbl 1344.60012
[12] Bourgade, P; Erdös, L; Yau, H-T, Universality of general \(β \)-ensembles, Duke Math. J., 163, 1127-1190, (2014) · Zbl 1298.15040
[13] Bourgade, P., Erdös, L., Yau, H.T.: Bulk Universality of general \(β \)-ensembles with non-convex potential. J. Math. Phys. 53(9), 095221, 19 (2012) · Zbl 1278.82032
[14] Braides, A.: \(Γ \)-Convergence for Beginners. Oxford University Press, Oxford (2002) · Zbl 1198.49001
[15] Brascamp, H.J., Lieb, E.H.: Functional Integration and its Applications. Clarendon Press, Oxford (1975) · Zbl 0348.26011
[16] Brauchart, JS; Hardin, DP; Saff, EB, Discrete energy asymptotics on a Riemannian circle, Uniform Distrib. Theory, 7, 77-108, (2012) · Zbl 1324.31006
[17] Caffarelli, LA; Roquejoffre, J-M; Sire, Y, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12, 1151-1179, (2010) · Zbl 1221.35453
[18] Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. In: Courant Lecture Notes in Mathematics. AMS, Providence (1999) · Zbl 0997.47033
[19] Deift, P., Gioev, D.: Random matrix theory: invariant ensembles and universality. In: Courant Lecture Notes in Mathematics. AMS, Providence (2009) · Zbl 1171.15023
[20] Dumitriu, I; Edelman, A, Matrix models for beta ensembles, J. Math. Phys., 43, 5830-5847, (2002) · Zbl 1060.82020
[21] Dyson, F.: Statistical theory of the energy levels of a complex system. Part I. J. Math. Phys. 3, 140-156 (1962). Part II, ibid. 157-165; Part III, ibid. 166-175 · Zbl 0118.23503
[22] Erdös, L; Schlein, B; Yau, HT, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab., 37, 815-852, (2009) · Zbl 1175.15028
[23] Forrester, P.J.: Log-gases and random matrices. In: London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010) · Zbl 1217.82003
[24] Johansson, K, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91, 151-204, (1998) · Zbl 1039.82504
[25] König, W, Orthogonal polynomial ensembles in probability theory, Probab. Surv., 2, 385-447, (2005) · Zbl 1189.60024
[26] Leblé, T.: A Uniqueness result for minimizers of the 1D log-gas renormalized energy. arXiv:1408.2283 · Zbl 0281.60091
[27] Leblé, T., Serfaty, S.: Large Deviation Principle for Empirical Fields of Log and Riesz Gases (in preparation) · Zbl 1397.82007
[28] Lenard, A, Exact statistical mechanics of a one-dimensional system with Coulomb forces, J. Math. Phys., 2, 682-693, (1961) · Zbl 0104.44604
[29] Lenard, A, Exact statistical mechanics of a one-dimensional system with Coulomb forces. III. statistics of the electric field, J. Math. Phys., 4, 533-543, (1963) · Zbl 0118.23503
[30] Mehta, M.L.: Random Matrices, 3rd edn. Elsevier/Academic Press, Amsterdam (2004) · Zbl 1107.15019
[31] Molchanov, SA; Ostrovski, E, Symmetric stable processes as traces of degenerate diffusion processes, Theory Probab. Appl., 14, 128-131, (1969) · Zbl 0281.60091
[32] Petrache, M., Serfaty, S.: Next order asymptotics and renormalized energy for Riesz interactions. arXiv:1409.7534 · Zbl 1373.82013
[33] Rougerie, N., Serfaty, S.: Higher dimensional Coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. (to appear, 2014) · Zbl 1338.82043
[34] Saff, E., Totik, V.: Logarithmic Potentials with External Fields. Springer, Springer (1997) · Zbl 0881.31001
[35] Sandier, E; Serfaty, S, From the Ginzburg-Landau model to vortex lattice problems, Commun. Math. Phys., 313, 635-743, (2012) · Zbl 1252.35034
[36] Sandier, E., Serfaty, S.: 2D Coulomb gases and the renormalized energy. Ann. Probab. (to appear, 2014) · Zbl 1328.82006
[37] Sandier, E., Serfaty, S.: Global minimizers for the Ginzburg-Landau functional below the first critical magnetic Field. Annales Inst. H. Poincaré, Anal. non linéaire 17(1), 119-145 (2000) · Zbl 0947.49004
[38] Shcherbina, M, Orthogonal and symplectic matrix models: universality and other properties, Commun. Math. Phys., 307, 761-790, (2011) · Zbl 1232.15027
[39] Shcherbina, M, Fluctuations of linear eigenvalue statistics of \(β \) matrix models in the multi-cut regime, J. Stat. Phys., 151, 1004-1034, (2013) · Zbl 1273.15042
[40] Shcherbina, M.: Change of variables as a method to study general \(β \)-models: bulk universality. arXiv:1310.7835 · Zbl 1296.82032
[41] Serfaty, S., Tice, I.: Lorentz space estimates for the Coulombian renormalized energy. Commun. Contemp. Math. 14(4), 1250027, 23 (2012) · Zbl 1255.35061
[42] Simon, B.: The Christoffel-Darboux kernel. In: Perspectives in PDE, Harmonic Analysis and Applications, a volume in honor of V.G. Maz’ya’s 70th birthday. Proceedings of Symposia in Pure Mathematics, vol. 79, pp. 295-335 (2008)
[43] Valkó, B; Virág, B, Continuum limits of random matrices and the Brownian carousel, Invent. Math., 177, 463-508, (2009) · Zbl 1204.60012
[44] Wigner, E, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math., 62, 548-564, (1955) · Zbl 0067.08403
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.