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The Brownian Cactus I. Scaling limits of discrete cactuses · Around

2011

The Brownian Cactus I. Scaling limits of discrete cactuses

Curien, Nicolas, Gall, Jean-François Le, Miermont, Grégory

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The cactus of a pointed graph is a discrete tree associated with this graph.

Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$.
We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense.
Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.

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Tutte, W.T
Original
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1993
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1993
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Le Gall, J.F
1999
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Burago, D., Burago, Y., Ivanov, S
2001
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Gromov, M
2001
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