Tunneling of the Kawasaki dynamics at low temperatures in two dimensions
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, p. 59-88

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature β on a two dimensional torus 𝛬 L ={0,,L-1} 2 . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are n 2 particles, n<L/2, and that the initial state is the configuration in which all sites of the square {0,,n-1} 2 are occupied. We show that in the time scale e 2β the process evolves as a Markov process on 𝛬 L which jumps from any site 𝐱 to any other site 𝐲𝐱 at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.

On considère un gaz sur réseau évoluant selon la dynamique de Kawasaki à température inverse β sur le tore bi-dimensionel 𝛬 L ={0,,L-1} 2 . Nous étudions l’évolution du processus parmi les états d’énergie minimale. Supposons la présence de n 2 particules, n<L/2 et qu’á l’état initial les sites du carré {0,,n-1} 2 soient tous occupés. Nous montrons qu’á l’échelle de temps e 2β le processus évolue comme une chaîne de Markov sur 𝛬 L qui saute d’un site 𝐱 vers un site 𝐲𝐱 à un taux strictement positif qui peut-être exprimé en terme de probabilités d’atteinte de dynamiques markoviennes élémentaires.

DOI : https://doi.org/10.1214/13-AIHP568
Classification:  82C44,  82C22,  60K35
Keywords: metastability, tunneling, lattice gases, kawasaki dynamics, capacities
@article{AIHPB_2015__51_1_59_0,
     author = {Beltr\'an, J. and Landim, Claudio},
     title = {Tunneling of the Kawasaki dynamics at low temperatures in two dimensions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {59-88},
     doi = {10.1214/13-AIHP568},
     zbl = {06412898},
     mrnumber = {3300964},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_1_59_0}
}
Beltrán, J.; Landim, C. Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 1, pp. 59-88. doi : 10.1214/13-AIHP568. http://www.numdam.org/item/AIHPB_2015__51_1_59_0/

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