Abstract
In this chapter, we solve the loop equations (Tutte’s equations), we compute explicitly the generating functions counting maps of given genus and boundaries. We are first going to solve them for planar maps with one boundary (the disk, i.e. planar rooted maps), then two boundaries (the cylinder), and then arbitrary genus and arbitrary number of boundaries. The disk case (planar rooted maps) was already done by Tutte [83–85]. Generating functions for higher topologies have been computed more recently [5, 31].
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Notes
- 1.
For simplicity, we assume that V ″(X i ) ≠ 0. The lemma remains true when V ″(X i ) = 0 but for the proof, one needs to go further in the Taylor expansion…
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Eynard, B. (2016). Solution of Tutte-Loop Equations. In: Counting Surfaces. Progress in Mathematical Physics, vol 70. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8797-6_3
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DOI: https://doi.org/10.1007/978-3-7643-8797-6_3
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