## Electronic Communications in Probability

### Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures

#### Abstract

The concept of metastate measures on the states of a random spin system was introduced to be able to treat the large-volume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strong-coupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 95, 12 pp.

Dates
Accepted: 27 November 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.ecp/1545102491

Digital Object Identifier
doi:10.1214/18-ECP200

Mathematical Reviews number (MathSciNet)
MR3896833

Zentralblatt MATH identifier
07023481

#### Citation

Cotar, Codina; Jahnel, Benedikt; Külske, Christof. Extremal decomposition for random Gibbs measures: from general metastates to metastates on extremal random Gibbs measures. Electron. Commun. Probab. 23 (2018), paper no. 95, 12 pp. doi:10.1214/18-ECP200. https://projecteuclid.org/euclid.ecp/1545102491

#### References

• [ADNS10] Louis-Pierre Arguin, Michael Damron, Charles Newman, and Daniel Stein. Uniqueness of ground states for short-range spin glasses in the half-plane. Comm. Math. Phys., 300(3):641–657, 2010.
• [ANS18] Louis-Pierre Arguin, Charles Newman, and Daniel Stein. A relation between disorder chaos and incongruent states in spin glasses on $\mathbb Z^d$. arXiv preprint arXiv:1803.02308, 2018.
• [AP18] Michael Aizenman and Ron Peled. A power-law upper bound on the correlations in the 2d random field Ising model. arXiv preprint arXiv:1808.08351, 2018.
• [ASZ18] Gérard Ben Arous, Eliran Subag, and Ofer Zeitouni. Geometry and temperature chaos in mixed spherical spin glasses at low temperature-the perturbative regime. arXiv preprint arXiv:1804.10573, 2018.
• [AW90] Michael Aizenman and Jan Wehr. Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys., 130(3):489–528, 1990.
• [BK88] Jean Bricmont and Antti Kupiainen. Phase transition in the 3d random field Ising model. Comm. Math. Phys., 116(4):539–572, 1988.
• [BK07] Marek Biskup and Roman Kotecký. Phase coexistence of gradient Gibbs states. Probab. Theory Related Fields, 139(1-2):1–39, 2007.
• [Bov06] Anton Bovier. Statistical mechanics of disordered systems: a mathematical perspective, volume 18. Cambridge University Press, 2006.
• [CH10] Jean-René Chazottes and Michael Hochman. On the zero-temperature limit of Gibbs states. Comm. Math. Phys., 297(1):265–281, 2010.
• [CK12] Codina Cotar and Christof Külske. Existence of random gradient states. Ann. Appl. Probab., 22(4):1650–1692, 2012.
• [CK15] Codina Cotar and Christof Külske. Uniqueness of gradient Gibbs measures with disorder. Probab. Theory Related Fields, 162(3-4):587–635, 2015.
• [CRL15] Daniel Coronel and Juan Rivera-Letelier. Sensitive dependence of Gibbs measures at low temperatures. J. Stat. Phys., 160(6):1658–1683, 2015.
• [Föl75] Hans Föllmer. Phase transition and Martin boundary. In Séminaire de Probabilités IX Université de Strasbourg, volume 465, pages 305–317, Berlin, Heidelberg, 1975.
• [FS97] Tadahisa Funaki and Herbert Spohn. Motion by mean curvature from the Ginzburg-Landau $\nabla \phi$ interface model. Comm. Math. Phys., 185(1):1–36, 1997.
• [Geo11] Hans-Otto Georgii. Gibbs measures and phase transitions, volume 9. Walter de Gruyter & Co., Berlin, second edition, 2011.
• [IK10] Giulio Iacobelli and Christof Külske. Metastates in finite-type mean-field models: visibility, invisibility, and random restoration of symmetry. J. Stat. Phys., 140(1):27–55, 2010.
• [Kle13] Achim Klenke. Probability theory: a comprehensive course. Universitext. Springer London, 2013.
• [Kül97] Christof Külske. Metastates in disordered mean-field models: random field and Hopfield models. J. Stat. Phys., 88(5-6):1257–1293, 1997.
• [NS97] Charles Newman and Daniel Stein. Metastate approach to thermodynamic chaos. Phys. Rev. E (3), 55(5, part A):5194–5211, 1997.
• [NS98] Charles Newman and Daniel Stein. Thermodynamic chaos and the structure of short-range spin glasses. In Mathematical aspects of spin glasses and neural networks, volume 41 of Progr. Probab., pages 243–287. Birkha̋user Boston, Boston, MA, 1998.
• [NS13] Charles Newman and Daniel Stein. Spin glasses and complexity. Princeton University Press, 2013.
• [vEK08] Aernout van Enter and Christof Külske. Nonexistence of random gradient Gibbs measures in continuous interface models in $d=2$. Ann. Appl. Probab., 18(1):109–119, 2008.
• [vER07] Aernout van Enter and Wioletta Ruszel. Chaotic temperature dependence at zero temperature. J. Stat. Phys., 127(3):567–573, 2007.
• [WMK15] Wenlong Wang, Jonathan Machta, and Helmut Katzgraber. Chaos in spin glasses revealed through thermal boundary conditions. Physical Review B, 92(9):094410, 2015.