Higher genus meanders and Masur–Veech volumes

A meander is a pair consisting of a straight line in the plane and of a smooth closed curve transversally intersecting the line, considered up to an isotopy preserving the straight line. The number of meanders with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> intersections grows exponentially with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , but asymptotics still remains conjectural. A meander defines a pair of transversally intersecting simple closed curves on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -sphere. In this paper we consider such pairs on a closed oriented surface of arbitrary genus. The number of these higher genus meanders still admits exponential upper and lower bounds as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> grows. Fixing the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> meanders with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> bigons and with at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> intersections and show that it grows polynomially with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> . We obtain a similar result in the case of oriented curves.