Explicit positive representation for weights on $R^d$

TL;DR

This paper explores methods to derive positive probability representations for complex weights in high-dimensional spaces, aiming to improve complex Langevin simulations, and demonstrates successful examples despite the underdetermined nature of the problem.

Contribution

It introduces a moment-matching approach to construct positive representations of complex weights, providing new examples and insights into overcoming limitations of complex Langevin methods.

Findings

Successful construction of positive representations in simple examples

The problem of deriving positive measures is highly underdetermined

Method offers potential improvements for complex Langevin simulations

Abstract

It is an old idea to replace averages of observables with respect to a complex weight by expectation values with respect to a genuine probability measure on complexified space. This is precisely what one would like to get from complex Langevin simulations. Unfortunately, these fail in many cases of physical interest. We will describe method of deriving positive representations by matching of moments and show simple examples of successful constructions. It will be seen that the problem is greatly underdetermined.