## Electronic Journal of Probability

### One-dimensional random field Kac's model: weak large deviations principle

#### Abstract

We present a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernouilli random variables. The results are valid for values of the temperature and magnitude of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minimizers. We give an explicit representation of the large deviation rate function and characterize its minimizers. We show that they are step functions taking two values, the two absolute minimizers of the free energy of the random Curie Weiss model. The points of discontinuity are described by a stationary renewal process related to the $h$-extrema of a bilateral Brownian motion studied by Neveu and Pitman, where $h$ depends on the temperature and magnitude of the random field. Our result is a complete characterization of the typical profiles of RFKM (the ground states) which was initiated in [2] and extended in [4].

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 47, 1372-1416.

Dates
Accepted: 16 June 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819508

Digital Object Identifier
doi:10.1214/EJP.v14-662

Mathematical Reviews number (MathSciNet)
MR2511287

Zentralblatt MATH identifier
1191.60117

Rights

#### Citation

Picco, Pierre; Orlandi, Enza. One-dimensional random field Kac's model: weak large deviations principle. Electron. J. Probab. 14 (2009), paper no. 47, 1372--1416. doi:10.1214/EJP.v14-662. https://projecteuclid.org/euclid.ejp/1464819508

#### References

• Asmussen, Søren. Applied probability and queues. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. x+318 pp. ISBN: 0-471-91173-9
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• Cassandro, Marzio; Orlandi, Enza; Presutti, Errico. Interfaces and typical Gibbs configurations for one-dimensional Kac potentials. Probab. Theory Related Fields 96 (1993), no. 1, 57–96.
• Cassandro, Marzio; Orlandi, Enza; Picco, Pierre. Typical configurations for one-dimensional random field Kac model. Ann. Probab. 27 (1999), no. 3, 1414–1467.
• Cassandro, Marzio; Orlandi, Enza; Picco, Pierre. The optimal interface profile for a non-local model of phase separation. Nonlinearity 15 (2002), no. 5, 1621–1651.
• Cassandro, Marzio; Orlandi, Enza; Picco, Pierre; Eulalia Vares, Maria. One-dimensional random field Kac's model: localization of the phases. Electron. J. Probab. 10 (2005), no. 24, 786–864 (electronic).
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
• Golosov, A. O. Limit distributions for a random walk in a critical one-dimensional random environment. (Russian) Uspekhi Mat. Nauk 41 (1986), no. 2(248), 189–190.
• Kac, M.; Uhlenbeck, G. E.; Hemmer, P. C. On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model.J. Mathematical Phys. 4 1963 216–228.
• Uhlenbeck, G. E.; Hemmer, P. C.; Kac, M. On the van der Waals theory of the vapor-liquid equilibrium.II. Discussion of the distribution functions.J. Mathematical Phys. 4 1963 229–247.
• Hemmer, P. C.; Kac, M.; Uhlenbeck, G. E. On the van der Waals theory of the vapor-liquid equilibrium. III. Discussion of the critical region. J. Mathematical Phys. 5 1964 60–74.
• Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
• Kesten, Harry.The limit distribution of SinaÄ­'s random walk in random environment.Phys. A 138 (1986), no. 1-2, 299–309.
• Lebowitz, J. L.; Penrose, O. Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition. J. Mathematical Phys. 7 1966 98–113.
• Neveu, J.; Pitman, J. Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. Séminaire de Probabilités, XXIII, 239–247, Lecture Notes in Math., 1372, Springer, Berlin, 1989.
• O. Penrose and J.L. Lebowitz. (1987). Towards a rigorous molecular theory of metastability.} Fluctuation Phenomena (W. Montroll and J.L. Lebowitz ed) North-Holland Physics Publishing.
• Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1991. x+533 pp. ISBN: 3-540-52167-4
• SinaÄ­, Ya. G. The limit behavior of a one-dimensional random walk in a random environment. (Russian) Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 247–258.
• Skorohod, A. V. Limit theorems for stochastic processes. (Russian) Teor. Veroyatnost. i Primenen. 1 (1956), 289–319.
• Taylor, Howard M. A stopped Brownian motion formula. Ann. Probability 3 (1975), 234–246.
• Williams, David. On a stopped Brownian motion formula of H. M. Taylor. Séminaire de Probabilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), pp. 235–239. Lecture Notes in Math., Vol. 511, Springer, Berlin, 1976.