An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.
Bijective counting of tree-rooted maps and shuffles of parenthesis systems
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abstract
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
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2019 1verdicts
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Random surfaces and Liouville quantum gravity
An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.