Uniform random sampling of simple branched coverings of the sphere by itself

We present the first polynomial uniform random sampling algorithm for simple branched coverings of the sphere by itself of degree n. More precisely, our algorithm generates in linear time increasing quadrangulations, which are equivalent combinatorial structures. Our result is based on the identification of some canonical labelled spanning trees, and yields a constructive proof of a celebrated formula of Hurwitz for the number of some factorizations of permutations in transpositions. The previous approaches were either non constructive or lead to exponential time algorithms for the sampling problem. Branched coverings of the sphere are 2-dimensional topological structures that have raised a lot of interest ever since the work of Hurwitz at the end of the 19th century. For instance, Okounkov and Pandharipande [17] have used these objects to derive an alternative to Kontsevitch’s proof of Witten’s celebrated conjecture. More recently, their relations to intersection numbers of moduli spaces and integrable hierachies as studied in mathematical physics have suggested that large random simple branched coverings provide an alternative model of discrete 2-dimensional pure quantum geometry (see e.g. [21] for a relatively accessible exposition). Our aim in the present article is to provide means to effectively sample these alternative random geometries, but since our approach is purely combinatorial we trade the topological definition of branched coverings for Hurwitz fundamental combinatorial representation (see however the appendix, and the complete and elegant treatment in [12]). Define a factorization in transpositions of the identity permutation idn on {1, . . . , n} to be a m-uple of transpositions τ1, . . . , τm such that τm · · · τ1 = idn. It is transitive if the graph Gτ on {1, . . . , n} with m edges given by the τi is connected, and minimal if m = 2n− 2. It can be checked that indeed this is the minimum number of transpositions in a transitive factorization of idn. Theorem 1 (Hurwitz (1891)). Simple branched coverings of the sphere by itself of degree n are encoded up to homeomorphisms of the domain by minimal transitive factorizations in transpositions of the identity of Sn, and their number, called n-th Hurwitz number, is n n−3(2n− 2)!. The usual model of quantum geometries is the uniform distribution on fixed size unlabelled planar quadrangulations, which was first studied analytically [3] and via Markov chain simulations [2]. Only later has it become possible to perform rigourous exact simulations via efficient (linear time) perfect random sampling [19, 18, 9]. The algorithmic technics underlying these samplers, mainly the identification of carefully chosen canonical spanning plane trees, have in turn triggered important progresses in the comprehension of the intrinsic geometries of random unlabelled quadrangulations [7], culminating with the construction of their continuum limit, the Brownian map [13, 15, 14]. We show here that a similar approach can be undertaken for simple branched coverings of the sphere: starting from a variant of the standard representation of factorizations as graphs embedded on surfaces, we first recast the problem in terms of some increasing quadrangulations. We then show that these labelled quadrangulations, which do not fit in the earlier framework, can be decomposed using labelled trees (akin to Cayley trees) instead of plane trees. We so obtain the first constructive proof of Hurwitz formula. Previous proofs were either non constructive (via differential equation hierachies [16], geometric considerations [11], or matrix integrals [5]) or yield exponential generation