Log-Sobolev和能隙的迁移原理及其在保守自旋系统中的应用
Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems
摘要：
当目标空间的曲率有界时，我们得到了将log-Sobolev不等式和Spectral-Gap不等式从源度量测度空间转移到目标度量测度空间的新原理.作为我们的主要应用，我们得到了各种保守自旋系统模型的log-Sobolev和Spectral-Gap常数的显式估计，该模型由非相互作用和弱相互作用粒子组成，约束为守恒的平均自旋。当自相互作用是强凸势的扰动时，这部分地恢复和部分地推广了Caputo，Chafa\"south，Grunewald，Landim，Lu，Menz，Otto，Panizo，Villani，Westdickenberg和Yau的结果.当自相互作用仅假设为（非强）凸的，在双边指数测度的情况下，我们得到尖锐的估计系统的谱隙作为平均自旋的函数，独立于系统的大小。
英文摘要：
We obtain new principles for transferring log-Sobolev and Spectral-Gap inequalities from a source metric-measure space to a target one, when the curvature of the target space is bounded from below. As our main application, we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of various conservative spin system models, consisting of non-interacting and weakly-interacting particles, constrained to conserve the mean-spin. When the self-interaction is a perturbation of a strongly convex potential, this partially recovers and partially extends previous results of Caputo, Chafa\"ı, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg and Yau. When the self-interaction is only assumed to be (non-strongly) convex, as in the case of the two-sided exponential measure, we obtain sharp estimates on the system's spectral-gap as a function of the mean-spin, independently of the size of the system.
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