Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

Beltrán, J.; Landim, C.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 59-88.

On considère un gaz sur réseau évoluant selon la dynamique de Kawasaki à température inverse $\beta$ sur le tore bi-dimensionel ${𝛬}_L={\{0,\cdots ,L-1\}}^2$. 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 ${\{0,\cdots ,n-1\}}^2$ soient tous occupés. Nous montrons qu’á l’échelle de temps ${\mathrm{e}}^{2\beta }$ le processus évolue comme une chaîne de Markov sur ${𝛬}_L$ qui saute d’un site $𝐱$ vers un site $𝐲\ne 𝐱$ à un taux strictement positif qui peut-être exprimé en terme de probabilités d’atteinte de dynamiques markoviennes élémentaires.

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $\beta$ on a two dimensional torus ${𝛬}_L={\{0,\cdots ,L-1\}}^2$. 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 ${\{0,\cdots ,n-1\}}^2$ are occupied. We show that in the time scale ${\mathrm{e}}^{2\beta }$ the process evolves as a Markov process on ${𝛬}_L$ which jumps from any site $𝐱$ to any other site $𝐲\ne 𝐱$ at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.

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