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Relaxation Time Of Anisotropic Simple Exclusion Processes And Quantum Heisenberg Models
, 2002
"... Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with N particles in a rectangle of Z . Every particle at row h tries to jump to an arbitrary empty site at row h ..."
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Cited by 16 (2 self)
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Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with N particles in a rectangle of Z . Every particle at row h tries to jump to an arbitrary empty site at row h 1 with rate q 1 q 2 (0; 1) is a measure of the drift driving the particles towards the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in N and in the size of the rectangle. The proof is inspired by a recent interesting technique envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for the non linear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spinS, S 2 N, XXZ chain and for the 111 interface of the spinS XXZ Heisenberg model, thus generalizing previous results valid only for spin 2 .
"Zero" temperature stochastic 3D Ising model and Dimer covering fluctuation: a first step towards interface mean curvature motion
 COMM. PURE APPL. MATH
, 2011
"... We consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disappears with ..."
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Cited by 10 (5 self)
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We consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disappears within a time τ+ which is at most L 2 (log L) c and at least L 2 /(c log L), for some c> 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large timescales, the evolution of the interface between “+ ” and “− ” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior τ+ ≃ const × L 2, conjectured on heuristic grounds [13, 7]. In dimension d = 2, τ+ can be shown to be of order L 2 without logarithmic corrections: the upper bound was proven in [8] and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [3].
CONVERGENCE TO EQUILIBRIUM OF BIASED PLANE PARTITIONS
, 903
"... Abstract. We study a singleflip dynamics for the monotone surface in (2 + 1) dimensions obtained from a boxed plane partition. The surface is analyzed as a system of nonintersecting simple paths. When the flips have a nonzero bias we prove that there is a positive spectral gap uniformly in the bo ..."
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Cited by 3 (2 self)
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Abstract. We study a singleflip dynamics for the monotone surface in (2 + 1) dimensions obtained from a boxed plane partition. The surface is analyzed as a system of nonintersecting simple paths. When the flips have a nonzero bias we prove that there is a positive spectral gap uniformly in the boundary conditions and in the size of the system. Under the same assumptions, for a system of size M, the mixing time is shown to be of order M up to logarithmic corrections.
The Spectral Gap for the
, 2001
"... We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆. Our main theorem is that there is a nonvanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about ..."
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We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain with Ising anisotropy ∆. Our main theorem is that there is a nonvanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large ∆, about its overall behavior as a function of ∆ and its dependence on the ground state, about the scaling of the gap and the structure of the lowlying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum. By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a SolidonSolid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.