Fast estimator of Jacobians in the Monte Carlo integration on Lefschetz thimbles

A solution to the sign problem is the so-called "Lefschetz thimble approach" where the domain of integration for field variables in the path integral is deformed from the real axis to a submanifold in the complex space. For properly chosen submanifolds ("thimbles") the sign problem disappears or is drastically alleviated. The parametrization of the thimble by real coordinates requires the calculation of a Jacobian with a computational cost of order O(V-3), where V is proportional to the spacetime volume. In this paper we propose two estimators for this Jacobian with a computational cost of order O(V). We discuss analytically the regimes where we expect the estimator to work and show numerical examples in two different models.