We consider a new functional inequality controlling the rate of relative entropy decay for random walks, the interchange process and more general block-type dynamics for permutations. The inequality lies between the classical logarithmic Sobolev inequality and the modified logarithmic Sobolev inequality, roughly interpolating between the two as the size of the blocks grows. Our results suggest that the new inequality may have some advantages with respect to the latter well known inequalities when multi-particle processes are considered. We prove a strong form of tensorization for independent particles interacting through synchronous updates. Moreover, for block dynamics on permutations we compute the optimal constants in all mean field settings, namely whenever the rate of update of a block depends only on the size of the block. Along the way we establish the independence of the spectral gap on the number of particles for these mean field processes. As an application of our entropy inequalities we prove a new subadditivity estimate for permutations, which implies a sharp upper bound on the permanent of arbitrary matrices with nonnegative entries, thus resolving a well known conjecture.
翻译:我们考虑一种新的功能不平等,以控制随机行走、交换过程和更一般的区块类型动态的相对衰变速度。这种不平等存在于古典对数 Sobolev 不平等和修改的对数 Sobolev 不平等之间,随着区块大小的扩大,两者大致相互交错。我们的结果表明,在考虑多粒子过程时,新的不平等对于后一种众所周知的不平等可能具有一定的优势。我们证明独立粒子通过同步更新进行互动的强强推形式。此外,对于所有平均字段环境中的区块动态,即当区块更新速度仅取决于区块大小时,我们计算最佳常数。在我们为这些中值场进程确定光谱差距对粒子数量的独立性的道路上,我们证明,在应用多粒子不平等时,新的多粒子估计值是新的次相加性估计值,这意味着任意矩阵与非内嵌条目的永久高度绑定,从而解决一个众所周知的推测。