Compact Brownian surfaces I: Brownian disks

By Jérémie Bettinelli, Grégory Miermont. 2015

104 citations recorded by the source corpus.

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family $$(\mathrm {BD}_L,\, 0<L<\infty )$$(BDL,0<L<∞) of random metric spaces homeomorphic to the closed unit disk of $$\mathbb {R}^2$$R2, the space $$\mathrm {BD}_L$$BDL being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where $$L=0$$L=0. Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/895abfd586f03d313f91b0540ba927aec4bc2cc4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding (2016, background): “alence of the QLE(8=3;0) metric on a unit boundary length p 8=3-quantum disk [DMS14] and the random metric disk with boundary called the Brownian disk. The Brownian disk is dened in dierent ways in [BM17] and [MS15a] and is further explored in [LGA18]. The equivalence of the Brownian disk denitions ”