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Register a new userOn double Hurwitz numbers with completed cycles
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In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection theory of some analogues of the moduli space of curves. The structure of these conjectural intersection numbers is discussed in detail.
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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 free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x leftrightarrow y$ 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 symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. 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 $(lambda_1,ldots,lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $lambda$ above $infty$ and $(2,ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.
A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.
Let the bipartite Turan number $ex(m,n,H)$ of a graph $H$ be the maximum number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and $n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$ for any positive integers $m,n,t$ with $ngeq mgeq tgeq frac{m}{2}+1$. This confirms the rest of a conjecture of Gy{o}ri cite{G97} (in a stronger form), and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma cite{JM18} for this range. We also prove a tight edge condition for consecutive even cycles in bipartite graphs, which settles a conjecture in cite{A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we obtain an estimate on the upper bound of the number of edges which are incident to at most one vertex in $C$. Our two results generalize or sharpen a classical theorem due to Jackson cite{J85} in different ways.
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)leq R_k(P_n)leq R_k(C_n)leq kn+o(n)$. The upper bound was recently improved by Sarkozy who showed that $R_k(C_n)leqleft(k-frac{k}{16k^3+1}right)n+o(n)$. Here we show $R_k(C_n) leq (k-frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
We study a $b$-deformation of monotone Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. It is a special case of the $b$-deformed weighted Hurwitz numbers recently introduced by the last two authors and has an interpretation in terms of generalized branched coverings of the sphere by non-oriented surfaces. We give an evolution (cut-and-join) equation for this model and we derive, by a method of independent interest, explicit Virasoro constraints from it, for arbitrary values of the deformation parameter $b$. We apply them to prove a conjecture of Feray on Jack characters. We also provide a combinatorial model of non-oriented monotone Hurwitz maps, which generalizes monotone transposition factorizations. In the case $b=1$ we show that the model obeys the BKP hierarchy of Kac and Van de Leur. As a consequence of our analysis we prove a recent conjecture of Oliveira and Novaes relating zonal polynomials with the dimensions of irreducible representations of $O(N)$. We also relate the model to an $O(N)$ version of the Brezin-Gross-Witten integral, which we solve explicitly in terms of Pfaffians in the case of even multiplicities.
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