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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.

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
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References:

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