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16
Mixing times of the biased card shuffling and the asymmetric exclusion process
 Trans. Amer. Math. Soc
, 2005
"... Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a “shuffle ” consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin ..."
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Cited by 22 (1 self)
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Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a “shuffle ” consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lowernumbered card comes before the highernumbered card. If the coin comes up tails, then we arrange the cards with the highernumbered card first. In this paper we prove that for all p � = 1/2, the mixing time of this card shuffling is O(N 2), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process. 1.
Fabio Hydrodynamic limit of a disordered lattice gas
 Probab. Theory Related Fields 127 (2003
"... ABSTRACT. We consider a model of lattice gas dynamics in Z d in the presence of disorder. If the particle interaction is only mutual exclusion and if the disorder field is given by i.i.d. bounded random variables, we prove the almost sure existence of the hydrodynamical limit in dimension d ≥ 3. The ..."
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Cited by 15 (4 self)
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ABSTRACT. We consider a model of lattice gas dynamics in Z d in the presence of disorder. If the particle interaction is only mutual exclusion and if the disorder field is given by i.i.d. bounded random variables, we prove the almost sure existence of the hydrodynamical limit in dimension d ≥ 3. The limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle.
Uniform Poincaré Inequalities For Unbounded Conservative Spin Systems: The NonInteracting Case
 Appl
"... We prove a uniform Poincaré inequality for noninteracting unbounded spin systems with a conservation law, when the singlesite potential is a bounded perturbation of a convex function. The result is then applied to GinzburgLandau processes to show diffusive scaling of the associated spectral gap. ..."
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Cited by 11 (1 self)
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We prove a uniform Poincaré inequality for noninteracting unbounded spin systems with a conservation law, when the singlesite potential is a bounded perturbation of a convex function. The result is then applied to GinzburgLandau processes to show diffusive scaling of the associated spectral gap.
Ferromagnetic ordering of energy levels
 J. of Stat. Phys
"... Dedicated to Elliott Lieb on the occasion of his seventieth birthday Abstract We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the LiebMattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest ..."
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Cited by 10 (4 self)
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Dedicated to Elliott Lieb on the occasion of his seventieth birthday Abstract We study a natural conjecture regarding ferromagnetic ordering of energy levels in the Heisenberg model which complements the LiebMattis Theorem of 1962 for antiferromagnets: for ferromagnetic Heisenberg models the lowest energies in each subspace of fixed total spin are strictly ordered according to the total spin, with the lowest, i.e., the ground state, belonging to the maximal total spin subspace. Our main result is a proof of this conjecture for the spin1/2 Heisenberg XXX and XXZ ferromagnets in one dimension. Our proof has two main ingredients. The first is an extension of a result of Koma and Nachtergaele which shows that monotonicity as a function of the total spin follows from the monotonicity of the ground state energy in each total spin subspace as a function of the length of the chain. For the second part of the proof we use the TemperleyLieb algebra to calculate, in a suitable basis, the matrix elements of the Hamiltonian restricted to each subspace of the highest weight vectors with a given total spin. We then show that the positivity properties of these matrix elements imply the necessary monotonicity in the volume. Our method also shows that the first excited state of the XXX ferromagnet on any finite tree has one less than maximal total spin.
Isolated eigenvalues of the ferromagnetic spinJ XXZ chain with kink boundary conditions
, 2008
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On
, 2002
"... the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations ..."
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the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations
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, 2002
"... the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations ..."
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the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations
On
, 2002
"... the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations ..."
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the dynamics of interfaces in the ferromagnetic XXZ chain under weak perturbations
Quantum Spin Systems after DLS1978
, 2005
"... In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Néel order at positive temperature for the spinS Heisenberg antiferromagnet on the ddimensional hypercubic lattice when either S ≥ 1 and d ≥ 3 or S = 1/2 and d is sufficiently large. This was the first proof of spontaneous b ..."
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In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Néel order at positive temperature for the spinS Heisenberg antiferromagnet on the ddimensional hypercubic lattice when either S ≥ 1 and d ≥ 3 or S = 1/2 and d is sufficiently large. This was the first proof of spontaneous breaking of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.