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期刊论文
OPTIMAL COUPLINGS AND APPLICATION TO RIEMANNIAN GEOMETRY
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The talk begins with some backgrounds of our study: The spectral gap for four classes of reversible Markov processes and the relation between the spectral gap and the phase transitions. Then, we introduce two aspects of our recent progress: 1) The estimates of the spectral gap (or the first non-trivial eigenvalue) of Laplacian on compact Riemannian manifold. 2) Optimal Markovian couplings. These explain the precise meaning of the vague title. The resulting estimates are quite unexpected, not only recover the known sharp estimates but also produce some new ones without using anything from the previous proofs. The optimal estimates come from the optimal couplings, which are often out of our probabilistic intuition. It seems to the author that the study of couplings is renewed but there is still a lot to be done. We emphasize the ideas, including the applications of the coupling technique, in terms of some simple examples. It is hoped that the materials presented here could be helpful not only for experts but also for newcomers.
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