Free functional inequalities on the circle

Abstract

In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities.These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far. arXiv:1511.05274v3 [math.PR] 23 Oct 2017for any smooth function φ on the interval [−2, 2]. This is further refined in [7] for the case of the quadratic potential V (x) = x 2 /2. The purpose of this paper is to expand the study of free functional inequalities to the circle. Previously, such inequalities were first introduced by Hiai, Petz and Ueda in [14]. Particularly, the transportation and Log-Sobolev inequalities on the circle were proved using unitary random matrix ensembles. We should point out a key fact which will play an important role in the sequel. In the random matrix approach from [14], the key idea is to look at the Haar measure on U (n), the space of unitary matrices. The main issue with this is that U (n) has Ricci curvature constant in all directions, except one direction in which it is 0. Therefore, the Bakry-Émery condition which is typically used to get functional inequalities does not hold in this case. The fix is to actually look at the subgroup of matrices with determinant one. This has the Ricci curvature constant and thus, we can apply the classical inequalities, which in the limit provides the free analogs. This phenomena produces various corrections as we will discuss below.Our approach is not based on the random matrix approach, but rather on direct methods based on mass transport tools. We recapture the free transportation and Log-Sobolev alongside with HWI and Brunn-Minkowski. We also introduce the free Poincaré inequality.We start the discussion on our approach introducing first the free Poincaré inequality (which to our knowledge is new) similar to (1.4). If µ is a probability on the unit circle, we say it satisfies a free Poincaré inequality with constant ρ > 0 ifholds for any smooth function f on the circle. Here we denote by α the Haar measure on the circle. Notice that there are similarities and also differences to the classical analog and the free version (1.4). A difference from the classical counterpart is due to the fact that the left hand sides of (1.4) and (1.5) are independent of the measure µ. From a random matrix perspective the fact that the left hand side of (1.4) and (1.5) do not depend on the measure µ is a reflection of universality in random matrix theory for the fluctuations (see the argument in [18] where (1.4) is introduced). The other particularity of (1.5) is that as opposed to the (1.4) and the classical Poincaré inequality, the right hand side has an extra term, namely f dµ 2 . In the case of...

Statistical mechanics of the periodic Benjamin–Ono equation

“…Then by [24,Corollary 2], the corresponding equilibrium density is continuous and satisfies ρ 0 (θ) > δ2 log 2 for all θ ∈ T. Popescu [24,Theorem 4] has also obtained a version of the free Poincaré inequality…”