## Final Report Summary - SAMTFP (Stochastic Analysis, Mass Transportation and Free Probability)

This project has a number of significant results obtained during the four years of support. They cover a number of important and diverse topics, from stochastic analysis in connection with geometric and analytic problems to random matrices and functional inequalities, particularly in free probability. These topics were well represented in the published papers all of them in prestigious journal which will certainly have an impact in their fields. Below is a short description of the achievements in each relevant publication.

One is “Analyticity of Planar Limits of a Matrix Model” by the researcher in collaboration with Stavros Garoufalidis and published in Annales Henri Poincare, April 2013, volume 14, no. 3, pp 499-565. The topic at hand revolves around a conjecture of ’t ‘Hoofts about the radius of convergence of the planar limit associated to a random matrix model. What is done there is starting with a formal random matrix model and the associated generating function of planar maps. Then, from an analytic perspective, there is a nice analysis of the equilibrium measure and the corresponding minimization of the logarithmic potential with external fields which is performed in terms of Chebyshev polynomials. Using this approach several very useful formulae were obtained for the generating function of the planar diagrams under various counting conditions. The other part consists in actually computing exactly some of these generating functions. The radius of convergence of the planar map is computed in terms of radius of convergence of the potential and it is proved that this is actually the best possible.

The next paper is “The One Dimensional Free Poincare Inequality” which is joint work of the researcher with Michel Ledoux and published in Transaction of AMS vol. 365 (2013), pp. 4811-4849. The main theme here is the natural analog of the Poincare inequality in free probability. Log-Sobolev and transportation inequalities were already introduced in the literature and a version of the Poincare inequality was discussed by Biane. Another version of the Poincare inequality was proved by the researcher with M. Ledoux in a previous publication which was suggested by the random matrix heuristics. The purpose of the present paper is to actually clarify the status of the true Poincare inequality. In the classical case, the Log-Sobolev and the transportation inequalities both imply the Poincare inequality and the proof is relatively elementary. The same is carried over in free probability in this article, however the path from Log-Sobolev or transportation to Poincare is highly nontrivial. It is though showed that the true version of the Poincare which comes naturally as the linearization of the Log-Sobolev and transportation is the version the authors discovered in their previous work.

Another paper is "Local One Dimensional Free Inequalities" published in Journal of Functional Analysis, volume 264, no. 6, March 2013, pp.1456--1479 which is an investigation of functional inequalities in free probability on the real line. This development is based on grounds laid down in the previous publication. What is proposed here is a version of the transportation inequality on the real line where the Wasserstein distance W_2 is replaced by the weaker W_1 distance. Usually, the transportation inequality assumes some form of convexity for the potential, but in this case this is not required, instead the base metric W_2 is replaced by W_1 as proposed by M. Maida and E. Maurel-Segala. In the present paper a local version (on the interval [-2,2]) is proved with sharp constants and then extended to the whole real line by arguments from logarithmic potential and mass transportation. Alongside, there is also a discussion on a local version for the Log-Sobolev and HWI inequality which are consequences of an interesting isometric property of the Hilbert transform between the L^2 spaces of the semicircular and the arcsine. There are also two open problems which are highly non-trivial.

"A refinement of Brascamp-Lieb-Poincare inequality in one dimension" (to appear in Comptes Rendus Mathematique) is an interesting and simple extension of the Brascamp-Lieb-Poincare inequality which has the flavor of Houdre-Kagan extension of the classical Poincare inequality for the normal distribution. This simple argument covers also the classical result of Houdre-Kagan as a particular case and shows that extensions of the Brascamp-Lieb require non-trivial properties beyond the mere convexity of the potential. The approach in this note is based on an observation of Helffer which gives a proof of the Brascamp-Lieb and a clever iteration of this leads to the extension. This works only in one dimension, in several dimensions the algebra seems more intricate.

The previous note plays an important role in the economy of the other paper "Refinements of the One Dimensional Free Poincare Inequality" by Christian Houdre and the researcher whose goal is two folds. One is to get a free Brascamp-Lieb type inequality for the convex potentials in free probability and the other is to get a version of Houdre-Kagan for the semicircular. Even though this seems very much parallel to the classical case, it is not as straightforward. The main tools used here were developed by the researcher in previous work also part of this project in the first two papers discussed in this description. As a side note, random matrix theory is used to heuristically guess the main operators which play the key role in the proofs.

"A Stochastic Target Approach to Ricci Flow on Surfaces" (joint work of the researcher with Robert W. Neel) is a paper accepted (subject to revisions) in Annals of Probability which gives the first purely probabilistic proof in the literature of the fact that on compact surfaces of non-positive Euler characteristic, the normalized Ricci-flow converges to a metric of constant curvature. According to the reviewers of the paper this is going to be a very influential paper. What is done there is first setup a stochastic target problem which is then used to represent the Ricci flow. In particular this yields the uniqueness of the flow. Based on this, the article proceeds to the convergence result of the flow toward a metric of constant curvature. The main tool is coupling of time changed Brownian motion, first for the flow itself and then for the first derivative. To get convergence of the second order derivatives, the coupling technique is not sufficient, and a coupling of three time changed Brownian motions is introduced to handle this case. This technique of triple coupling does not appear in the literature except a few lines in a paper by Cranston for a simple case and seems to be a very promising tool in getting second order estimates, thus it is of independent interest and requires further investigation. For higher derivatives, the use of induction, the initial equation for the flow and another coupling does the rest. Overall this is a very long and technical paper with lots of ideas and potential for further development.

Also in stochastic analysis but on a different theme, the paper “Shy and Fixed-Distance Coupling on Riemaniann Manifolds” which is joint work of the researcher with Mihai N. Pascu develops two types of couplings of Brownian motions. The first one is shy coupling, a notion introduced by Benjamini-Burdzy-Chen and is a coupling which keeps the moving particles away from each other. One of the main statements of the ppaer is that on a Riemanian manifold with bounded sectional curvature and positive injectivity radius, there is a shy coupling of Brownian motions. More interestingly, if the Ricci curvature is nonnegative, the sectional curvature is bounded above and the injectivity radius is positive, then there is a coupling of two Brownian motions, X_t, Y_t for which the distance between X_t and Y_t is constant in time. As a side note this is sharpening a result of von Renese and Sturm which states that the Ricci curvature nonnegative implies that there exist a coupling of Brownian motions such that X_t and Y_t stay at distance at most the distance between X_0 and Y_0. As other potential applications of this result it one should point out the longstanding Hot Spots conjecture.

Other papers are either in a preprint form and near to submission or still in works. For instance one is on functional inequalities in free probability on the circle. Another is on optimal alignments with respect to two random scoring functions in collaboration with Henry Matzinger. Also work in progress is in the areas of first and second order freeness for Wigner matrices and some applications to matrices with dependencies.

To close this summary, these results will have a serious impact in the community. There is enough depth, richness and innovation which will generate enough interest in the years to come.