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...e π is the stationary distribution. Moreover, combining our reduction from the card shuffling to the exclusion process with the bound on the spectral gap of the (continuous time) exclusion process in =-=[4]-=-, it is straightforward to verify that there exists positive c1 and c2 such that indeed the spectral gap satisfies c1 ≤−λ2(N) ≤ c2 for all N. However, the probability space contains elements of very s...

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...ve result is the uniformity in the disorder configuration. Its proof is based on some clever technique developed recently in [13] to deal with the Kac model for the Boltzmann equation and extended in =-=[12]-=- and [11] to other kind of diffusions. For other models of lattice gas dynamics like the dilute Ising lattice gas in the Griffiths regime the above uniformity will no longer be available and a more so...

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...proof of Theorem 1.1 will be given in the next two sections. It relies on a simple but powerful idea recently introduced by Carlen, Carvalho and Loss in [9]. A similar technique was then used also in =-=[8]-=- to study the relaxation to equilibrium for a conservative lattice gas dynamics. The argument of [9] essentially shows that in view of the symmetry of the measures (1.2) one can reduce the problem to ...

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...en Markov processes and certain quantum spin models such as the XXX and XXZ, as well as closely related models, such as simple exclusion processes, has been fruitfully exploited in the past. E.g., in =-=[2]-=-, Caputo and Martinelli proved good lower bounds for the gap of the spin-S XXZ chain by mapping the problem to an asymmetric simple exclusion process and applying probabilistic techniques to the latte...

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...shed by Koma and Nachtergaele [11, 12] and Matsui [15]. The existence of a spectral gap for general j was proven by Koma, Nachtergaele and Starr [13]. Recently, it was proven by Caputo and Martinelli =-=[8]-=- that the gap for spin j scales asymptotically like j, however, no precise values are known. This verifies a conjecture stated in [13], and comes as a surprise since the spin-j Ising Key words and phr...

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...shed by Koma and Nachtergaele [11, 12] and Matsui [15]. The existence of a spectral gap for general j was proven by Koma, Nachtergaele and Starr [13]. Recently, it was proven by Caputo and Martinelli =-=[8]-=- that the gap for spin j scales asymptotically like j, however, no precise values are known. This verifies a conjecture stated in [13], and comes as a surprise since the spin-j Ising Key words and phr...

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...shed by Koma and Nachtergaele [12, 13] and Matsui [16]. The existence of a spectral gap for general j was proven by Koma, Nachtergaele and Starr [14]. Recently, it was proven by Caputo and Martinelli =-=[8]-=- that the gap for spin j scales asymptotically like j, however, no precise values are known. This verifies a conjecture stated in [14], and comes as a surprise since the spin-j Ising chain (which can ...

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...an the one provided by Theorem 7.3 by up to a factor of 2. By somewhat different but not unrelated methods excellent estimates were obtained by Caputo and Martinelli for the higher spin XXZ chains in =-=[13]-=-. They proved that the spectral gap is bounded above and below by quantities of the form constant times S, supporting the conjecture that the limit limS→∞ γ(S)/S exists and is strictly positive [43]. ...

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...operator ˜ h (r), defined below in (2.1), where are is the solution of the equation µ = with η determined by ∆ = cosh η. +∞∑ x=−∞ tanh(η(x − r)) A partial result towards this conjecture was proved in =-=[4]-=-. Namely, there it is shown that there are constants c1 > 0, and c2 > 0, independent of M and J, such that c1J ≤ γJ,M ≤ c2J . In this paper we prove that the value of the gap claimed in the conjecture...