Universal Scaling Limit Research Articles

Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

We show that the classical Szasz analytic function S N (f)(x) is obtained by applying the pseudo-differential operator f(N −1 D θ ) to the Bergman kernels for the Bargmann–Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric Kähler manifold, defining the generalized Szasz analytic function $S_{h^{N}}(f)(x)$ . About $S_{h^{N}}(f)(x)$ , we prove that it admits complete asymptotics and there exists a universal scaling limit. As an example, we will further compute $S_{h^{N}}(f)(x)$ for the Bergman metric on the unit ball.

Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulae for the bulk–boundary and boundary–boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian continuum random tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk–loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometryof triangles made of three geodesic paths joining them. We also study the geometry ofminimal separating loops, i.e. paths of minimal length among all closed paths passing byone of the three vertices and separating the two others in the quadrangulation. Weconcentrate on the universal scaling limit of large quadrangulations, also known as theBrownian map, where pairs of geodesic paths or minimal separating loops have commonparts of non-zero macroscopic length. This is the phenomenon of confluence, whichdistinguishes the geometry of random quadrangulations from that of smooth surfaces. Wecharacterize the universal probability distribution for the lengths of these commonparts.

Distribution laws for integrable eigenfunctions

On détermine l'asymptotique semi-classique des fonctions propres jointes de l'action d'un tore sur une variété kählérienne torique. Ces variétés sont des modèles de systèmes complètement intégrables en géométrie complexe. On démontre que les fonctions propres ressemblent ponctuellement à des gaussiennes centrées aux tores correspondants. De plus, on prouve qu'il existe une limite universelle gaussienne de la fonction de distribution renormalisée auprès de son centre, et on détermine sa distribution limite non-universelle loin de son centre.

EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds: An addendum

We define a Gaussian measure on the space H 0 J (M, L N ) of almost holomorphic sections of powers of an ample line bundle L over a symplectic manifold (M, ω), and calculate the joint probability densities of sections taking prescribed values and covariant derivatives at a finite number of points. We prove that they have a universal scaling limit as N → ∞. This result will be used in another paper to extend our previous work on universality of scaling limits of correlations between zeros to the almost-holomorphic setting.

Correlations Between Zeros and Supersymmetry

In our previous work [BSZ2], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions.

Power spectra of higher period multiplings in area-preserving maps

We have studied the power spectra of higher period p-multiplings for p = 3, 4, 5 and 6 in area-preserving maps. The ratio of the successive average heights of the peaks of the spectrum for each p-multiplying is found to approach a universal scaling limit that increases with p.