Ionel Popescu

The one dimensional free Poincaré inequality

In this paper we discuss the natural candidate for the one dimensional free Poincare inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincare is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincare inequality revolves around the counting number operator for the Chebyshev polynomials of the first kind with respect to the arcsine law on [−2, 2]. This counting number operator appears naturally in a representation of the minimum of the logarithmic energy with external fields discovered in Analyticity of the planar limit of a matrix model by S. Garoufalidis and the second author as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities. Classically, Poincare’s inequality for a probability measure μ on R states that there is a constant ρ > 0 such that for any compactly supported smooth function f , (0.1) ρVarμ(f) ≤ ∫ |∇f |dμ, with the notation Varμ(f) = ∫ f dμ− ( ∫ f dμ). This is in fact a statement about the spectral gap of the operator L, whose Dirichlet form is Γ(f, f) = ∫ |∇f | dμ (and invariant measure μ). This inequality is actually one member of a family of functional inequalities which are connected by implications among them. For example, among others, the transportation and Log-Sobolev inequalities always imply the Poincare with the same constant (see e.g. [1, 17, 4, 20]). With the boom in the interest of large dimensional phenomena, one natural question is to ask what happens with the functional inequalities in the limit. This was studied in various forms for various measures in infinite dimensions, as for example the Wiener measures with a few samples [11], [9], [21], [8], [14]. The important part in dealing with these infinite dimensional objects was due to the dimension independent constants in the finite dimensional approximations. Important interesting limiting objects are obtained in free probability by considering random matrices. It is well known that properly normalized, the eigenvalue distribution of the Gaussian Unitary Ensemble converges (in mean and almost surely) to the semicircular law. On the other hand, applying classical functional inequalities to the distribution of random matrices in dimension n and taking their limits, one obtains various functional inequalities for the semicircular law. This Received by the editors May 10, 2011 and, in revised form, October 21, 2011. 2010 Mathematics Subject Classification. Primary 46L54; Secondary 60B20, 60E15, 33D45. The second author was partially supported by Marie Curie Action grant no. 249200. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication 4811

A stochastic target approach to Ricci flow on surfaces

We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi’s representation of mean curvature flow. We prove a verification/uniqueness theorem, and then consider geometric consequences of this stochastic representation. Based on this stochastic approach, we give a proof that, for surfaces of nonpositive Euler characteristic, the normalized Ricci flow converges to a constant curvature metric exponentially quickly in every Ck-norm. In the case of C0 and C1-convergence, we achieve this by coupling two particles. To get C2-convergence (in particular, convergence of the curvature), we use a coupling of three particles. This triple coupling is developed here only for the case of constant curvature metrics on surfaces, though we suspect that some variants of this idea are applicable in other situations and therefore be of independent interest. Finally, for k≥3, the Ck-convergence follows relatively easily using induction and coupling of two particles. None of these techniques appear in the Ricci flow literature and thus provide an alternative approach to the field.

An upper bound on the convergence rate of a second functional in optimal sequence alignment

Consider finite sequences

Non-normal Limiting Distribution for Optimal Alignment Scores of Strings in Binary Alphabets

X_{[1,n]}=X_1\dots X_n

Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

and

General tridiagonal random matrix models, limiting distributions and fluctuations

Y_{[1,n]}=Y_1\dots Y_n

Analyticity of the Planar Limit of a Matrix Model

of length

Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

n

Local functional inequalities in one-dimensional free probability☆

, consisting of i.i.d.\ samples of random letters from a finite alphabet, and let

Shy and fixed-distance couplings of Brownian motions on manifolds

S