A Gaussian correlation inequality for plurisubharmonic functions

A geometric approach to correlation inequalities in the plane

The celebrated Four Functions Theorem of Ahlswede and Daykin is a functional correlation inequality on distributive lattices with myriad applications. Ruozzi proved a variant of the inequality and used it to settle a major conjecture in the area of graphical models. We prove a new functional correlation inequality in the same vein which simplifies the proof of both the Four Functions Theorem and of Ruozzi's inequality and suggests a unified picture for correlation inequalities on distributive lattices.

Local Inequalities for Multivariate Polynomials and Plurisubharmonic Functions

We show that Schwarz symmetrization does not increase the Monge-Ampere energy for S-1-invariant plurisubharmonic functions in the ball. As a result, we derive a sharp Moser-Trudinger inequality for such functions. We also show that similar results do not hold for other balanced domains except for complex ellipsoids, and discuss related questions for convex functions.

We consider a large class of models which share the essential features of the Kondo model. Bounds on the susceptibility of the impurity spin are derived as consequences of general inequalities for quantum correlation functions. We also obtain bounds for the spin polarization in the presence of an external field.

We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

Characterization of equality in the correlation inequality for convex functions, the U-conjecture

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to $e^{-|t|^p}$ and symmetric $p$-stable random variables, where $p\in(0,2]$. We obtain various sharp moment and entropy comparison estimates for weighted sums of independent Gaussian mixtures and investigate extensions of the B-inequality and the Gaussian correlation inequality in the context of Gaussian mixtures. We also obtain a correlation inequality for symmetric geodesically convex sets in the unit sphere equipped with the normalized surface area measure. We then apply these results to derive sharp constants in Khintchine inequalities for vectors uniformly distributed on the unit balls with respect to $p$-norms and provide short proofs to new and old comparison estimates for geometric parameters of sections and projections of such balls.

Nonsymmetric examples for Gaussian correlation inequalities

We present a Gaussian correlation inequality which is closely related to a result of Schechtman, Schlumprecht and Zinn (1998) on the well-known Gaussian correlation conjecture. The usefulness of the inequality is demonstrated by several important applications to the estimates of small ball probability.

The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of polynomial, algebraic and analytic functions. The paper has much in common with an earlier paper [Br] of the author.

On the Moser–Trudinger inequality in complex space

Moser-Trudinger inequality for the complex Monge-Ampère equation

A symmetrization inequality for plurisubharmonic functions

We show, apart from a correlation inequality for the 6-point function, that the critical or continuum limit of (φ^4)_4 lattice field theory is trivial, if some deviations from the mean field theory law are present at the critical point, and if we impose a mild behavior of bare 4-point coupling constant as we approach the critical point. These results are derived from the bounds on the renormalized coupling constant of lattice (φ^4)_d field theory.