Stable quadrangulations and stable spheres

TL;DR

This paper investigates the scaling limits of random quadrangulations derived from Bienaymé-Galton-Watson trees with heavy-tailed offspring distributions, revealing their Hausdorff dimension and proposing a new class called the alpha-stable sphere.

Contribution

It introduces the alpha-stable sphere as a candidate for the universal limit of these quadrangulations and studies their geometric and volume fluctuation properties.

Findings

Scaling limits have Hausdorff dimension 2α/(α-1).

Subsequential limits exist and are conjectured to be unique and spherical.

Volume fluctuations resemble those of stable trees.

Abstract

We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaym\'e-Galton-Watson trees with stably-decaying offspring tails with an exponent $\alpha$ in (1, 2). We show that these quadrangulations admit subsequential scaling limits wich all have Hausdorff dimension $\frac{2\alpha}{\alpha-1}$ almost surely. We conjecture that the limits are unique and spherical, and we introduce a candidate for the limit that we call the $\alpha$-stable sphere. In addition, we conduct a detailed study of volume fluctuations around typical points in the limiting maps, and show that the fluctuations share similar characteristics with those of stable trees.