Abstract
On a general separable state space, let P be a Markovian kernel, reversible with respect to a probability μ, such that there is a spectral gap for the operator Id—P in L 2(μ). Let P β be the Metropolis kernel associated to P and to a potential βU, where U is a measurable bounded function and β≥0 is the inverse temperature. We are interested in its spectral gap λ(β) and specially in the behaviour for large β of the quantity -β-1 In (λ(β)). By analogy with some classical cases, we have conjectured that it should converge to c, the largest secondary well exit height, but as we will see on a counter-example, this is not true, since -β-1 In (λ(β)) can asymptotically oscilate, and even if it converges, the limit can be different from c. We also get similar results for the asymptotical behaviour at small temperature of the isoperimetric constants I(β), but for them the study is a little more satisfactory, as we can conclude to the convergence of -β-1 In (λ(β)) to c, if it is equal to ess supμ U-essinfμ U.
Keywords
- Measurable Bounded Function
- Markovian Kernel
- Isoperimetric Constant
- Nous Allons
- Basse Temperature
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© 1998 Springer-Verlag
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Miclo, L. (1998). Trous spectraux à basse température: un contre-exemple à un comportement asymptotique escompté. In: Azéma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII. Lecture Notes in Mathematics, vol 1686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101749
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DOI: https://doi.org/10.1007/BFb0101749
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