Formation of infinite loops for an interacting bosonic loop soup

TL;DR

This paper analyzes the limiting measure of the Feynman loop representation of the Bose gas at high densities, revealing phase transitions and the emergence of random interlacements linked to Bose--Einstein condensation.

Contribution

It provides the first rigorous computation of the limiting measure for non mean-field energies and identifies phase transition phenomena involving random interlacements.

Findings

Limiting measure assigns positive weight to random interlacements at high densities

Critical density shifts compared to mean-field models

Discontinuity in interlacement density at the critical point

Abstract

We compute the limiting measure for the Feynman loop representation of the Bose gas for a non mean-field energy. As predicted in previous works, for high densities the limiting measure gives positive weight to random interlacements, indicating the quantum Bose--Einstein condensation. We prove that in many cases there is a shift in the critical density compared to the free/mean-field case, and that in these cases the density of the random interlacements has a jump-discontinuity at the critical point.