Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

DeepDyve requires Javascript to function. Please enable Javascript on your browser to continue.

Counting Surfaces

Part of the Progress in Mathematical Physics Book Series (volume 70)

DOI: 10.1007/978-3-7643-8797-6

ISBN: 978-3-7643-8796-9

Publisher: Springer Basel - Springer Journals

Mar 22, 2016

LitStream

Chapter 1Maps and Discrete Surfaces

Eynard, Bertrand

Pages: 1 –24

[In this chapter we introduce definitions of maps, which are discrete surfaces obtained by gluing polygons along their sides, and we define generating functions to count them. We also derive Tutte’s equations, which are recursive equations satisfied by the generating functions.]
LitStream

Chapter 2Formal Matrix Integrals

Eynard, Bertrand

Pages: 25 –51

[In this chapter we introduce the notion of a formal matrix integral, which is very useful for combinatorics, as it turns out to be identical to the generating function of maps of Chap. 1]
LitStream

Chapter 3Solution of Tutte-Loop Equations

Eynard, Bertrand

Pages: 53 –143

[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...
LitStream

Chapter 4Multicut Case

Eynard, Bertrand

Pages: 145 –168

[Readers only interested in the combinatorics of maps can easily skip this chapter and move directly to Chap. 5]
LitStream

Chapter 5Counting Large Maps

Eynard, Bertrand

Pages: 169 –236

[Initially, in quantum gravity and string theory, the problem of counting maps, i.e. surfaces made of polygons, was introduced only as a discretized approximation for counting continuous surfaces. The physical motivation is the following: in string theory, particles are 1-dimensional loops called...
LitStream

Chapter 6Counting Riemann Surfaces

Eynard, Bertrand

Pages: 237 –333

[In the previous chapter, we have computed the asymptotic generating functions of large maps, and we have seen that they are related to the ( p, q) minimal model.]
LitStream

Chapter 7Topological Recursion and Symplectic Invariants

Eynard, Bertrand

Pages: 335 –363

[We have seen, in almost all previous chapters, that symplectic invariants and topological recursion play an important role. They give the solution to Tutte’s recursion equation for maps, they give the formal expansion of various matrix integrals, including Kontsevich integral, and they also give...
LitStream

Chapter 8Ising Model

Eynard, Bertrand

Pages: 365 –407

[In statistical physics, the Ising model represents a simplified model for magnetization. Each piece of the surface (here each face of a map) carries a unit of magnetization, pointing either upward + or downward −. This can also be represented as a map with bicolored faces black/white, or + ∕−,...

results per page

Prev
1
Next

Browse All eBooks