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Formal matrix integrals and combinatorics of maps

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arxiv math-ph/0611087 v3 pith:LA2KU32J submitted 2006-11-30 math-ph hep-thmath.MP

Formal matrix integrals and combinatorics of maps

classification math-ph hep-thmath.MP
keywords matrixintegralsformalmapscombinatoricsconvergentsomewell
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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This article is a short review on the relationship between convergent matrix integrals, formal matrix integrals, and combinatorics of maps. We briefly summarize results developed over the last 30 years, as well as more recent discoveries. We recall that formal matrix integrals are identical to combinatorial generating functions for maps, and that formal matrix integrals are in general very different from convergent matrix integrals. Finally, we give a list of the classical matrix models which have played an important role in physics in the past decades. Some of them are now well understood, some are still difficult challenges.

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Cited by 2 Pith papers

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  1. A bijective topological recursion for maps

    math.GT 2026-07 conditional novelty 8.0

    Iterating Tutte's edge-erasing procedure until the topology changes yields a bijective pair-of-pants excision that reproduces (blobbed) topological recursion for maps and stuffed maps.

  2. A bijective topological recursion for maps

    math.GT 2026-07 conditional novelty 8.0

    A bijective skinning procedure yields the topological recursion for maps of arbitrary topology at the level of formal generating series, with no analyticity assumptions, including stuffed maps and a combinatorial recu...