D. , ALL-ORDER EXPANSION AND THE TOPOLOGICAL RECURSION of 2g non-contractible cycles such that A i ? B j = ? ij , A i ? A j = 0

, There exists a unique basis du 1 ,. .. , du g such that The ? (g) n satisfy many remarkable properties, one of which is that they are symmetric. This is not obvious from the definition because, The surface L has a g-dimensional vector space of holomorphic differential forms

. , On the right, an example of a 3-colored melonic graph

. , The second graph of Fig.1.1 with explicit indices

. , On the right: its splitting with the intermediate matrix field M (i) of color i (the image is just suggestive and does not contain any "hidden" meaning)

. , A standard tableau of shape, vol.3

G. Akemann, J. Baik, and P. D. Francesco, The Oxford handbook of random matrix theory, p.16, 2011.

A. , Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra, Comm. Math. Phys, vol.338, issue.1, p.62, 2015.

J. Ambjorn and L. O. Chekhov, A matrix model for hypergeometric Hurwitz numbers, Translation of Teoret. Mat. Fiz, vol.181, issue.3, pp.421-435, 2014.

J. Ambjorn, B. Durhuus, and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Modern Physics Letters A, vol.06, issue.12, p.23, 1991.

G. W. Anderson, A. Guionnet, and O. Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol.118, p.16, 2010.

R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc, vol.54, issue.319, p.83, 1985.

D. Bessis, C. Itzykson, and J. B. Zuber, Quantum field theory techniques in graphical enumeration, Advances in Applied Mathematics, vol.1, issue.2, p.16, 1980.

V. Bonzom and S. Dartois, Blobbed topological recursion for the quartic melonic tensor model, vol.39, p.111, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01833818

V. Bonzom, R. Gurau, and V. Rivasseau, Random tensor models in the large N limit: Uncoloring the colored tensor models, Phys. Rev. D, vol.85, p.84037, 1924.
URL : https://hal.archives-ouvertes.fr/hal-00816588

A. Borodin, On a family of symmetric rational functions, Adv. Math, vol.306, p.48, 2017.

A. Borodin and I. Corwin, Macdonald processes. Probab. Theory Related Fields, vol.158, pp.225-400, 2014.

G. Borot, Blobbed topological recursion, Theoretical and Mathematical Physics, vol.185, issue.3, p.111, 2015.

G. Borot, B. Eynard, M. Mulase, and B. Safnuk, A matrix model for simple Hurwitz numbers, and topological recursion, J. Geom. Phys, vol.61, issue.2, p.65, 2011.

G. Borot, B. Eynard, and A. Weisse, Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces, Selecta Math. (N.S.), vol.23, issue.2, p.16, 2017.
URL : https://hal.archives-ouvertes.fr/cea-01278006

G. Borot and A. Guionnet, Asymptotic expansion of ? matrix models in the one-cut regime, Comm. Math. Phys, vol.317, issue.2, p.19, 2013.

G. Borot, R. Kramer, D. Lewanski, A. Popolitov, and S. Shadrin, Special cases of the orbifold version of Zvonkine's r-ELSV formula, 1964.

G. Borot and S. Shadrin, Blobbed topological recursion: properties and applications, Mathematical Proceedings of the Cambridge Philosophical Society, vol.162, issue.1, p.16, 2017.

V. Bouchard and M. Mariño, Hurwitz numbers, matrix models and enumerative geometry, From Hodge Theory to Integrability and tQFT: tt*-geometry, Proc. Symp. Pure Math, 2007.

A. Buryak and R. J. Tessler, Matrix models and a proof of the open analog of Witten's conjecture, Comm. Math. Phys, vol.353, issue.3, p.16, 2017.

L. Cantini, A. Garbali, J. De-gier, and M. Wheeler, Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries, J. Phys. A, vol.49, issue.44, p.444002, 2016.

R. Cavalieri, P. Johnson, and H. Markwig, Tropical Hurwitz numbers, Journal of Algebraic Combinatorics, vol.32, issue.2, p.61, 2010.
DOI : 10.1017/cbo9781316569252.013
URL : https://link.springer.com/content/pdf/10.1007%2Fs10801-009-0213-0.pdf

T. Ceccherini-silberstein, F. Scarabotti, and F. Tolli, Representation theory of the symmetric groups: The Okounkov-Vershik approach, character formulas, and partition algebras, Cambridge Studies in Advanced Mathematics, vol.53, p.58, 2010.

L. Chekhov and B. Eynard, Hermitian matrix model free energy: Feynman graph technique for all genera, Journal of High Energy Physics, issue.03, p.14, 2006.

I. Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol.319, issue.9, 2005.
DOI : 10.1017/cbo9780511546501

P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant lecture notes in mathematics, p.20, 1999.

P. Deift, T. Kriecherbauer, K. T-r-mclaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Communications on Pure and Applied Mathematics, vol.52, issue.11, p.20, 1999.

P. D. Francesco, P. Ginsparg, and J. Zinn-justin, 2D gravity and random matrices, Phys. Rep, vol.254, issue.1-2, p.15, 1995.
DOI : 10.1016/0370-1573(94)00084-g
URL : http://arxiv.org/pdf/hep-th/9306153v2.pdf

E. Duchi, D. Poulalhon, and G. Schaeffer, Bijections for simple and double Hurwitz numbers, p.61, 2014.

T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves. Inventiones mathematicae, vol.146, p.63, 2001.

N. M. Ercolani, K. D. , and T. Mclaughlin, Asymptotics of the partition function for random matrices via riemann-hilbert techniques and applications to graphical enumeration, International Mathematics Research Notices, vol.18, issue.14, p.19, 2003.

B. Eynard, Counting Surfaces, p.16, 2016.
DOI : 10.1007/978-3-7643-8797-6

B. Eynard, M. Mulase, and B. Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, Publications of the Research Institute for Mathematical Sciences, vol.7, p.65, 2011.

B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics, vol.1, issue.2, p.23, 2007.
DOI : 10.4310/cntp.2007.v1.n2.a4
URL : https://hal.archives-ouvertes.fr/hal-00130963

B. Eynard and N. Orantin, Topological recursion in enumerative geometry and random matrices, Journal of Physics A: Mathematical and Theoretical, vol.42, issue.29, p.113, 2009.
DOI : 10.1088/1751-8113/42/29/293001

C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the ? g conjecture, Ann. of Math, vol.157, issue.2, p.74, 2003.

J. S. Frame, G. De-b.-robinson, and R. M. Thrall, The hook graphs of the symmetric groups, Canadian J. Math, vol.6, p.54, 1954.

I. P. Goulden, D. M. Jackson, and R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math, vol.198, issue.1, p.75, 2005.
DOI : 10.1016/j.aim.2005.01.008
URL : https://doi.org/10.1016/j.aim.2005.01.008

M. Guay-paquet and J. Harnad, 2D Toda ?-functions as combinatorial generating functions, Lett. Math. Phys, vol.105, issue.6, p.62, 2015.
DOI : 10.1007/s11005-015-0756-z

R. Gurau, Colored group field theory, Communications in Mathematical Physics, vol.304, issue.1, pp.69-93, 2011.
DOI : 10.1007/s00220-011-1226-9
URL : http://arxiv.org/pdf/0907.2582v1.pdf

R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré, vol.13, issue.3, p.27, 2012.

R. Gurau and J. P. Ryan, SIGMA Symmetry Integrability Geom, Methods Appl, vol.8, issue.020, 2012.

R. A. Gustafson, A generalization of Selberg's beta integral, Bull. Amer. Math. Soc. (N.S.), vol.22, issue.1, p.86, 1990.

J. Harnad, Quantum Hurwitz numbers and Macdonald polynomials, J. Math. Phys, vol.57, issue.11, p.62, 2016.

J. Harnad, A. Yu, and . Orlov, Hypergeometric tau functions, Hurwitz numbers and enumeration of paths, Communications in Mathematical Physics, vol.338, issue.1, p.62, 2015.

A. Hoshino, M. Noumi, and J. Shiraishi, Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for MacdonaldKoornwinder polynomials, Mosc. Math. J, vol.15, issue.2, p.95, 2015.

A. Hurwitz, Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann, vol.39, issue.1, p.60, 1891.

A. Hurwitz, Ueber die Anzahl der Riemann'schen Flächen mit gegebenen Verzweigungspunkten, Math. Ann, vol.55, issue.1, pp.53-66, 1901.

H. Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A, vol.69, issue.8, pp.1-18, 1970.

C. G. Jacobi, De functionibus alternantibus earumque divisione per productum e differentiis elementorum conflatum, J. Reine Angew. Math, vol.22, pp.360-371

P. Johnson, Double Hurwitz numbers via the infinite wedge, Transactions of the American Mathematical Society, vol.367, issue.9, p.103, 2015.

P. Johnson, R. Pandharipande, and H. Tseng, Abelian Hurwitz-Hodge integrals, Michigan Math. J, vol.60, issue.1, p.64, 2011.

A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Reports on Mathematical Physics, vol.5, issue.1, p.57, 1974.

V. G. Kac, Infinite-dimensional Lie algebras, 1990.

N. Kawanaka, A q-series identity involving Schur functions and related topics. Osaka, J. Math, vol.36, issue.1, p.52, 1999.

M. E. Kazarian and S. K. Lando, Combinatorial solutions to integrable hierarchies, Russian Mathematical Surveys, vol.70, issue.3, p.103, 2015.

S. Kerov and G. Olshanski, Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. I Math, vol.319, issue.2, p.66, 1994.

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix airy function, Comm. Math. Phys, vol.147, issue.1, p.64, 1992.

T. H. Koornwinder, Askey-Wilson polynomials for root systems of type BC, Hypergeometric functions on domains of positivity, Jack polynomials, and applications, vol.138, p.83, 1991.

S. Lando and A. K. Zvonkin, Graphs on surfaces and their applications, vol.16, p.108, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00307202

R. Langer, M. J. Schlosser, and S. O. Warnaar, Theta functions, elliptic hypergeometric series, and Kawanaka's Macdonald polynomial conjecture, SIGMA Symmetry Integrability Geom. Methods Appl, vol.5, p.52, 2009.

M. Lassalle, An explicit formula for the characters of the symmetric group, Math. Ann, vol.340, issue.2, p.56, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00622757

D. E. Littlewood, On certain symmetric functions, Proc. London Math. Soc, vol.11, issue.3, pp.485-498, 1961.

I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol.157, p.86, 2003.
DOI : 10.1017/cbo9780511542824
URL : http://www.numdam.org/article/SB_1994-1995__37__189_0.pdf

I. G. Macdonald, Oxford classic texts in the physical sciences, vol.47, p.50, 1998.

M. Mariño, Chern-Simons theory, matrix models and topological strings, International Series of Monographs on Physics, vol.131, p.15, 2005.

M. L. Mehta, Random matrices, Pure and Applied Mathematics, vol.142, p.15, 2004.

K. Mimachi, A duality of MacDonald-Koornwinder polynomials and its application to integral representations, Duke Math. J, vol.107, issue.2, p.84, 2001.

G. E. Murphy, A new construction of Young's seminormal representation of the symmetric groups, Journal of Algebra, vol.69, issue.2, p.57, 1981.

S. M. Natanzon, A. Yu, and . Orlov, BKP and projective Hurwitz numbers, Letters in Mathematical Physics, vol.107, issue.6, p.62, 2017.

V. A. Nguyen, Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles, vol.66, p.69, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01280165

V. A. Nguyen, S. Dartois, and B. Eynard, An analysis of the intermediate field theory of T4 tensor model, Journal of High Energy Physics, vol.2015, issue.1, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01093064

A. Okounkov, Toda equations for Hurwitz numbers, Mathematical Research Letters, vol.7, issue.4, p.61, 2000.

A. Yu, D. M. Orlov, and . Shcherbin, Hypergeometric solutions of soliton equations, Teoret. Mat. Fiz, vol.128, issue.1, p.105, 2001.

I. Pak, G. Panova, and E. Vallejo, Kronecker coefficients: the tensor square conjecture and unimodality, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), p.55, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01207577

E. M. Rains and S. O. Warnaar, Bounded Littlewood identities, vol.52, p.92, 2009.
DOI : 10.1016/j.jcta.2012.03.001
URL : https://doi.org/10.1016/j.jcta.2012.03.001

E. M. Rains, BC n-symmetric polynomials, Transform. Groups, vol.10, issue.1, p.86, 2005.

E. M. Rains, Elliptic analogues of the Macdonald and Koornwinder polynomials, Proceedings of the International Congress of Mathematicians, vol.IV, p.9, 2010.

E. M. Rains and M. Vazirani, Vanishing integrals of Macdonald and Koornwinder polynomials, Transform. Groups, vol.12, issue.4, pp.725-759, 2007.

V. Rivasseau, The tensor track, iii. Fortschritte der Physik, vol.62, p.27, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00912659

N. Sasakura, Tensor model for gravity and orientability of manifold, Modern Physics Letters A, vol.06, issue.28, p.23, 1991.

G. Schaeffer, Planar maps. Handbook of Enumerative Combinatorics, vol.87, p.61, 2015.
URL : https://hal.archives-ouvertes.fr/inria-00099359

S. Shadrin, L. Spitz, and D. Zvonkine, On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc, vol.86, issue.2, p.73, 2012.

S. Shadrin, L. Spitz, and D. Zvonkine, Equivalence of ELSV and Bouchard-Mariño conjectures for r-spin Hurwitz numbers, Math. Ann, vol.361, issue.3-4, p.67, 2015.

P. Sniady, Combinatorics of asymptotic representation theory, European Congress of Mathematics, p.55, 2013.

G. 't-hooft, A planar diagram theory for strong interactions, Nuclear Physics B, vol.72, issue.3, p.16, 1974.

J. F. Van-diejen, Commuting difference operators with polynomial eigenfunctions, Compos. Math, vol.95, issue.2, p.85, 1995.

A. M. Vershik and S. V. Kerov, Asymptotic theory of characters of the symmetric group, Functional Analysis and Its Applications, vol.15, issue.4, p.55, 1981.

S. O. Warnaar, Rogers-Szegö polynomials and Hall-Littlewood symmetric functions, Journal of Algebra, vol.303, issue.2, p.50, 2006.
DOI : 10.1016/j.jalgebra.2006.04.026
URL : https://doi.org/10.1016/j.jalgebra.2006.04.026

J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika, vol.20, issue.1, p.15, 1928.

E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry, vol.1, p.76, 1990.

R. Wong, Asymptotic approximations of integrals, Society for Industrial and Applied Mathematics, p.109, 2001.

J. Yao, S. Zheng, and Z. Bai, Large sample covariance matrices and highdimensional data analysis, of Cambridge Series in Statistical and Probabilistic Mathematics, vol.39, p.16, 2015.
DOI : 10.1017/cbo9781107588080