scholarly journals Fermionic Approach to Weighted Hurwitz Numbers and Topological Recursion

2017 ◽  
Vol 360 (2) ◽  
pp. 777-826 ◽  
Author(s):  
A. Alexandrov ◽  
G. Chapuy ◽  
B. Eynard ◽  
J. Harnad
Keyword(s):  
Hurwitz Numbers ◽  
Topological Recursion

scholarly journals Weighted Hurwitz Numbers and Topological Recursion

2020 ◽  
Vol 375 (1) ◽  
pp. 237-305 ◽  
Author(s):  
A. Alexandrov ◽  
G. Chapuy ◽  
B. Eynard ◽  
J. Harnad
Keyword(s):  
Hurwitz Numbers ◽  
Topological Recursion

scholarly journals Weighted Hurwitz numbers and topological recursion: An overview

2018 ◽  
Vol 59 (8) ◽  
pp. 081102 ◽  
Author(s):  
A. Alexandrov ◽  
G. Chapuy ◽  
B. Eynard ◽  
J. Harnad
Keyword(s):  
Hurwitz Numbers ◽  
Topological Recursion

scholarly journals Topological recursion and a quantum curve for monotone Hurwitz numbers

2017 ◽  
Vol 120 ◽  
pp. 19-36 ◽  
Author(s):  
Norman Do ◽  
Alastair Dyer ◽  
Daniel V. Mathews
Keyword(s):  
Hurwitz Numbers ◽  
Topological Recursion

scholarly journals A matrix model for simple Hurwitz numbers, and topological recursion

2011 ◽  
Vol 61 (2) ◽  
pp. 522-540 ◽  
Author(s):  
Gaëtan Borot ◽  
Bertrand Eynard ◽  
Motohico Mulase ◽  
Brad Safnuk
Keyword(s):  
Matrix Model ◽  
Hurwitz Numbers ◽  
Topological Recursion

scholarly journals Generating Functions of Bipartite Maps on Orientable Surfaces

10.37236/5511 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Guillaume Chapuy ◽  
Wenjie Fang
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Complex Analysis ◽  
Orientable Surface ◽  
Hurwitz Numbers ◽  
Change Of Variables ◽  
Topological Recursion ◽  
Number Of Faces ◽  
New Variables ◽  
Formal Power

We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L_g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $F_g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet /Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result  complements recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from Eynard's "topological recursion" that he applied in particular to even-faced maps (unconventionally called "bipartite maps" in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.

scholarly journals Simple Maps, Hurwitz Numbers, and Topological Recursion

2020 ◽  
Vol 380 (2) ◽  
pp. 581-654
Author(s):  
Gaëtan Borot ◽  
Elba Garcia-Failde
Keyword(s):  
Matrix Model ◽  
Hermitian Matrix ◽  
Universal Relation ◽  
Hurwitz Numbers ◽  
Gaussian Unitary Ensemble ◽  
Model Interpretation ◽  
Topological Recursion ◽  
Generating Series ◽  
Hermitian Matrix Model ◽  
Symplectic Invariance

Abstract We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and (higher order) free cumulants established by Collins et al. [22], and implement the symplectic transformation $$x \leftrightarrow y$$ x ↔ y on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y. We propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1-hermitian matrix model, which is believed to be true but has not been proved yet. Our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces—which are generated by the Gaussian Unitary Ensemble—and with boundary perimeters $$(\lambda _1,\ldots ,\lambda _n)$$ ( λ 1 , … , λ n ) are strictly monotone double Hurwitz numbers with ramifications $$\lambda $$ λ above $$\infty $$ ∞ and $$(2,\ldots ,2)$$ ( 2 , … , 2 ) above 0. Combining with a recent result of Dubrovin et al. [24], this implies an ELSV-like formula for these Hurwitz numbers.

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