An Explicit Formula for Witten’s 2-Correlators

In memory of Maryam Mirzakhani who inspired us in so many ways.Abstract. A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with 2N crossings grows exponentially when N grows, but the longstanding problem on the precise asymptotics is still out of reach.We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as N tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinders. The proofs combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics. √ 145 12of the corresponding critical exponent α in a two-dimensional CFT with central charge c = −4 coupled to gravity is given in [DiFGG2].We say that a meander has a maximal arc ("rainbow " in terminology of [ACPRS]) if it has an arc joining the leftmost and the rightmost crossings with the horizontal line. Otherwise the meander does not a have maximal arc. The meander on the left of Figure 1 has a maximal arc, while the one on the right does not.By minimal arc ("pimple" in terminology of [ACPRS], or "internal arch" in terminology of [DiFGG1]) we call an arc which does not have any crossings inside.

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“…Values of 2-correlators τ k τ 3g−1−k g can be obtained in a particularly efficient way through the following recursive relations found in [Zog1]:…”

Section: Proofs Of the Main Results For Fixed Values Of G And Nmentioning

confidence: 99%

“…In the context of meanders we need only 1-and 2-correlators. The 1-correlators admit a closed formula [Wi], and the 2-correlators admit a linear recursion [Zog1] and precise estimates [DGZZ3].…”

Section: Figure 1 Equivalent Meanders With 10 Crossingsmentioning

confidence: 99%

Enumeration of Meanders and Masur–veech Volumes

Delecroix

Goujard

Zograf

et al. 2020

Forum of Mathematics, Pi

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“…The inequality, where λ appears on the left-hand side for the first time, is the induction assumption applied to each term. The next inequality follows from the inequality λ(g, L 0 ) > λ(g, L 0 + 1), see (23). The equality which follows is the definition (13) of δ string 0 s+1 , k + 1, d 1 , .…”

Section: Remark 14mentioning

confidence: 91%

“…We performed a detailed analysis of ε(k, 3g − 1 − k) in [7] based on [23]. In particular, for large g the error term ε(k, 3g−1−k) rapidly tends to 0 when k approaches 3g−1 2 , so the statement of the above theorem can be seriously strengthened, if needed.…”

Section: State Of the Artmentioning

confidence: 99%

Uniform Lower Bound for Intersection Numbers of ψ-Classes

Delecroix

Goujard

Zograf

et al. 2020

SIGMA

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We approximate intersection numbers ψ d1 1 • • • ψ dn n g,n on Deligne-Mumford's moduli space M g,n of genus g stable complex curves with n marked points by certain closedform expressions in d 1 ,. .. , d n. Conjecturally, these approximations become asymptotically exact uniformly in d i when g → ∞ and n remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximating expressions multiplied by an explicit factor λ(g, n), which tends to 1 when g → ∞ and d 1 + • • • + d n−2 = o(g).

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“…Remark 4. Formula (51) with n = 2 is given in [4], which is shown in [20] to be equivalent to Zograf's formula [31] for the 2-point intersection numbers.…”

Section: Psi-class Intersection Numbers and Proof Of Theoremmentioning

confidence: 99%