## Electronic Journal of Probability

### Equilibrium Fluctuations for a One-Dimensional Interface in the Solid on Solid Approximation

Gustavo Posta

#### Abstract

An unbounded one-dimensional solid-on-solid model with integer heights is studied. Unbounded here means that there is no a priori restrictions on the discrete gradient of the interface. The interaction Hamiltonian of the interface is given by a finite range part, proportional to the sum of height differences, plus a part of exponentially decaying long range potentials. The evolution of the interface is a reversible Markov process. We prove that if this system is started in the center of a box of size $L$ after a time of order $L^3$ it reaches, with a very large probability, the top or the bottom of the box.

#### Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 29, 962-987.

Dates
Accepted: 18 July 2005
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464816830

Digital Object Identifier
doi:10.1214/EJP.v10-270

Mathematical Reviews number (MathSciNet)
MR2164036

Zentralblatt MATH identifier
1109.60086

Rights

#### Citation

Posta, Gustavo. Equilibrium Fluctuations for a One-Dimensional Interface in the Solid on Solid Approximation. Electron. J. Probab. 10 (2005), paper no. 29, 962--987. doi:10.1214/EJP.v10-270. https://projecteuclid.org/euclid.ejp/1464816830

#### References

• P. Diaconis and L. Saloff-Coste. Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 (1993), no. 3, 696-730.
• R. Dobrushin, R. Koteck? and S. Shlosman. Wulff Construction. A Global Shape from Local Interaction. Translation of Mathematical Monographs, 104 (1992). AMS.
• G. F. Lawler and A. D. Sokal. Bounds on the $L^2$ Spectrum for Markov Chains and Markov Processes: a Generalization of Cheeger's Inequality. Trans. Amer. Math. Soc. 309 (1988), no. 2, 557-580.
• T. M. Liggett. Interacting Particles Systems. Grundlehren der Mathematischen Wissenschaften 276 (1985). Springer-Verlag, New York-Berlin.
• S. T. Lu and H.-T. Yau. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Comm. Math. Phys. 156 (1993), 399-433.
• F. Martinelli. On the two dimensional dynamical Ising model in the phase coexistence region. J. Statist. Phys. 76 (1994), no. 5-6, 1179-1246.
• F. Martinelli. Lectures on Glauber dynamics for discrete spin models in Lectures on probability theory and statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 (1999) 93-191. Sringer-Verlag.
• F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region I: the attractive case. Comm. Math. Phys. 161 (1994), no. 3, 447-486.
• F. Martinelli and E. Olivieri. Approach to equilibrium of glauber dynamics in the one phase region II: the general case. Comm. Math. Phys. 161 (1994), no. 3, 487-514.
• G. Posta. Spectral Gap for an Unrestricted Kawasaki Type Dynamics, ESAIM Probability & Statistics 1 (1997), 145-181.
• A. D. Sokal and L. E. Thomas. Absence of mass gap for a class of stochastic contour models. J. Statist. Phys. 51 (1988), no. 5-6, 907-947.
• D. W. Stroock and B. Zegarlinski. The logarithmic Sobolev inequality for discrete spin on a lattice. Comm. Math. Phys. 149 (1992), no. 1, 175-193.