Demo · Dual sampling
How do you turn the flip of a coin into a random variable whose expected value is a scattering amplitude? By evaluating the curve integral with Monte Carlo — and sampling cleverly, with a tropical bias, directly on the surface.
The curve-integral formula writes the amplitude as a single integral over the surface — no diagrams. But it lives in many dimensions. How do you actually evaluate it?
Turn the flip of a coin into a random variable whose average is the amplitude. Draw points at random, evaluate the integrand, take the mean — plain Monte Carlo.
The catch: the integrand is spiky, concentrated near the surface's degenerations. Uniform random points almost always land in the flats and miss the peaks — the estimate barely converges.
Tropical importance sampling. Use the Newton-polytope / tropical shadow of the integrand to draw samples where it is large. The piecewise-linear tents hug the peaks, and the variance collapses.
What makes it scale: drop the Feynman bias that an amplitude is a sum over diagrams, and re-arrange the sum the way a dual triangulation of the curve integral suggests — realized as a random process directly on the surface.
Run it for massive tr φ³ in two dimensions, up to ten loops. The running average settles onto the amplitude using far fewer sample points than there are Feynman diagrams.
A scattering amplitude becomes something you can estimate, like the area under a curve, even when writing it as a sum of diagrams is hopeless. The sampler is a stochastic process on surfaces, so its cost tracks the geometry rather than the diagram count. The paper closes with an idea borrowed from machine learning: parametrize a family of samplers and learn the best one by solving a convex optimization problem.