Compact Brownian Surfaces II

Orientable Surfaces

von Jérémie Bettinelli, Grégory Miermont

Hardcover

Kurzbeschreibung des Verlags

Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with n faces and boundary component lengths of order \sqrt n or of lower order. Endow this quadrangulation with the usual graph metric renormalized by n^{-1/4}, mark it on each boundary component, and endow it with the counting measure on its vertex set renormalized by n^{-1}, as well as the counting measure on each boundary component renormalized by n^{-1/2}. We show that, as n goes to infinity, this random marked measured metric space converges in distribution for the Gromov–Hausdorff–Prokhorov topology, toward a random limiting marked measured metric space called a Brownian surface .

This extends known convergence results of uniform random planar quadrangulations with at most one boundary component toward the Brownian sphere and toward the Brownian disk , by considering the case of quadrangulations on general compact orientable surfaces. Our approach consists in cutting a Brownian surface into elementary pieces that are naturally related to the Brownian sphere and the Brownian disk and their noncompact analogs, the Brownian plane and the Brownian half-plane, and to prove convergence results for these elementary pieces, which are of independent interest.

• Reihe: Memoirs of the European Mathematical Society
• Themen: Mathematik und Naturwissenschaften Mathematik Wahrscheinlichkeitsrechnung und Statistik
• ISBN: 9783985471027
• Sprache: Englisch
• Erscheinungsdatum: 01.03.2026
• Größe: 24 x 17 cm
• Verlag: EMS Press