##
**Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus.**
*(English)*
Zbl 1326.60137

This article is concerned with a continuous-time Markov chain, which is parameterized by inverse temperature \(\beta\) and is defined on the state space \(\{-1, 1\}^{T_L}\), where \(T_L\) is the vertex set of the two-dimensional torus of side \(2L+1\). The evolution is described by an Ising gas subjected to the Kawasaki dynamics, i.e., each pair of adjacent vertices exchange their spin values at random times. Given the state of the process \(\eta\), transitions occur at a rate \(\exp ( - \beta ( H_{x, y} (\eta) - H (\eta)_+ ) )\), where \(H (\eta)_+ \) is the positive part of the Hamiltonian and \(H_{x, y} (\eta)\) is the Hamiltonian when the vertices \(x\) and \(y\) have exchanged their spins; alternatively, one can envision that the particle at \(x\) has jumped at \(y\) and vice versa. Applying a versatile method by J. Beltrán and the second author for the description of the asymptotic dynamics of one-parameter families of processes [J. Stat. Phys. 140, No. 6, 1065–1114 (2010; Zbl 1223.60061); ibid. 149, No. 4, 598–618 (2012; Zbl 1260.82063)], the authors of the paper under review prove three asymptotic theorems whenever the process is continuous at the zero-temperature limit. In all theorems, the initial state is assumed to be a box configuration, i.e., a box in the torus whose vertices have the same spin and the rest of the vertices have the other spin. Both sides of the torus and of a box configuration are assumed to depend on \(\beta\). The theorems hold for large \(L\) and as \(\beta\) tends to infinity. The first theorem states that, under an assumption that allows to discard all particle jumps of lower order in \(\beta\) and another assumption, the trace of the process, appropriately speeded up and normalized, converges (in the Skorokhod topology) to a Brownian motion. The second theorem provides a sufficient condition for the process visiting almost surely certain small subsets of the state space, for all times of the order at most the speed-up factor encountered in the first theorem. According to the third theorem, under certain assumptions that include the ones of the latter two theorems, the center of mass of the state of the process, appropriately speeded up and normalized, converges (in the Skorokhod topology) to a Brownian motion. From the three theorems, one concludes that “in a proper time scale the particles form almost always a square and that the center of mass of the square evolves as a Brownian motion when the temperature vanishes.” In terms of the Ising model literature, this is a statement on the evolution of the condensate. Lower and upper bounds of the order of the time scale are also proved in the article.

Reviewer: Stylianos Scarlatos (Thessaloniki)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60J27 | Continuous-time Markov processes on discrete state spaces |

60J65 | Brownian motion |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

### Keywords:

Ising model; Kawasaki dynamics; continuous-time Markov chain; zero-temperature limit; scaling limit; absorption; Brownian motion
PDF
BibTeX
XML
Cite

\textit{B. Gois} and \textit{C. Landim}, Ann. Probab. 43, No. 4, 2151--2203 (2015; Zbl 1326.60137)

### References:

[1] | Beltrán, J. and Landim, C. (2010). Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140 1065-1114. · Zbl 1223.60061 |

[2] | Beltrán, J. and Landim, C. (2011). Metastability of reversible finite state Markov processes. Stochastic Process. Appl. 121 1633-1677. · Zbl 1223.60060 |

[3] | Beltrán, J. and Landim, C. (2012). Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Related Fields 152 781-807. · Zbl 1251.60070 |

[4] | Beltrán, J. and Landim, C. (2012). Tunneling and metastability of continuous time Markov chains II, the nonreversible case. J. Stat. Phys. 149 598-618. · Zbl 1260.82063 |

[5] | Beltrán, J. and Landim, C. (2015). Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. Ann. Inst. Henri Poincaré Probab. Stat. 51 59-88. · Zbl 1314.82036 |

[6] | Bovier, A., den Hollander, F. and Nardi, F. R. (2006). Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary. Probab. Theory Related Fields 135 265-310. · Zbl 1099.60066 |

[7] | Bovier, A., den Hollander, F. and Spitoni, C. (2010). Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures. Ann. Probab. 38 661-713. · Zbl 1193.60114 |

[8] | den Hollander, F., Nardi, F. R., Olivieri, E. and Scoppola, E. (2003). Droplet growth for three-dimensional Kawasaki dynamics. Probab. Theory Related Fields 125 153-194. · Zbl 1040.60083 |

[9] | den Hollander, F., Olivieri, E. and Scoppola, E. (2000). Metastability and nucleation for conservative dynamics. J. Math. Phys. 41 1424-1498. · Zbl 0977.82030 |

[10] | Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22 . Mathematical Association of America, Washington, DC. · Zbl 0583.60065 |

[11] | Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049 |

[12] | Funaki, T. (2004). Zero temperature limit for interacting Brownian particles. I. Motion of a single body. Ann. Probab. 32 1201-1227. · Zbl 1121.82028 |

[13] | Funaki, T. (2004). Zero temperature limit for interacting Brownian particles. II. Coagulation in one dimension. Ann. Probab. 32 1228-1246. · Zbl 1122.82029 |

[14] | Gaudillière, A. (2009). Condenser physics applied to Markov chains: A brief introduction to potential theory. Available at . |

[15] | Gaudillière, A., den Hollander, F., Nardi, F. R., Olivieri, E. and Scoppola, E. (2009). Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics. Stochastic Process. Appl. 119 737-774. · Zbl 1158.60043 |

[16] | Gaudillière, A., Olivieri, E. and Scoppola, E. (2005). Nucleation pattern at low temperature for local Kawasaki dynamics in two dimensions. Markov Process. Related Fields 11 553-628. · Zbl 1109.82017 |

[17] | Jara, M., Landim, C. and Teixeira, A. (2011). Quenched scaling limits of trap models. Ann. Probab. 39 176-223. · Zbl 1211.60040 |

[18] | Jara, M., Landim, C. and Teixeira, A. (2014). Universality of trap models in the ergodic time scale. Ann. Probab. 42 2947-2557. · Zbl 1309.60098 |

[19] | Landim, C. (2014). Metastability for a non-reversible dynamics: The evolution of the condensate in totally asymmetric zero range processes. Comm. Math. Phys. 330 1-32. · Zbl 1305.82045 |

[20] | Lawler, G. F. (1991). Intersections of Random Walks . Birkhäuser, Boston, MA. · Zbl 1228.60004 |

[21] | Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times . Amer. Math. Soc., Providence, RI. · Zbl 1160.60001 |

[22] | Nardi, F. R., Olivieri, E. and Scoppola, E. (2005). Anisotropy effects in nucleation for conservative dynamics. J. Stat. Phys. 119 539-595. · Zbl 1073.82026 |

[23] | Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications 100 . Cambridge Univ. Press, Cambridge. |

[24] | Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65 117-149. · Zbl 0060.46001 |

[25] | Peierls, R. (1936). Statistical theory of adsorption with interaction between the adsorbed atoms. Proc. Camb. Phil. Soc. 32 471. · Zbl 0015.09003 |

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.