Hurwitz numbers, ribbon graphs, and tropicalization

@article{Johnson2012HurwitzNR,
  title={Hurwitz numbers, ribbon graphs, and tropicalization},
  author={P. Johnson},
  journal={arXiv: Algebraic Geometry},
  year={2012}
}
  • P. Johnson
  • Published 6 March 2013
  • Mathematics
  • arXiv: Algebraic Geometry
The double Hurwitz number Hg(µ, ν) has at least four equivalent definitions. Most naturally, it counts the covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain collections of permutations. More recently, it has been shown to be equivalent to a count of certain ribbon graphs, or as a weighted count of certain labeled graphs. This note is an expository account of the equivalences between these definitions… 

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References

SHOWING 1-10 OF 10 REFERENCES
Towards the geometry of double Hurwitz numbers
The Moduli Space of Curves, Double Hurwitz Numbers, and Faber’s Intersection Number Conjecture
We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of $${{\mathbb{P}_1}}$$ with given ramification
Double Hurwitz numbers via the infinite wedge
We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula in the work of Goulden, Jackson, and Vakil for one
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only
KP hierarchy for Hodge integrals
An algebro-geometric proof of Witten's conjecture
We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating
Principles of Algebraic Geometry
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications
Arnol′d, Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges, Funktsional
  • Anal. i Prilozhen
  • 1996
Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves