Formation of infinite loops for an interacting bosonic loop soup

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.

Figures (6)

• Figure 1: The density of the random interlacements/condensate (red, ) and the finite loops/bulk (blue, ) for various models in certain dimensions.
• Figure 2: The density of the random interlacements/condensate () and the finite loops/bulk () for the partial HYL model in different dimensions and different parameters.
• Figure 3: Sketch of the behaviour the thermodynamic functions relation density and chemical potential. The behaviour for $d\geq5$ is drawn as a solid line. For the first derivatives, the behaviour is qualitatively different for $d=3,4$ and is drawn as a blue dotted line.
• Figure 4: Two sketches of $I$, once for $\mu<0$ (black, solid line) and once for $\mu=0$ (blue, dotted).
• Figure 5: Sketch of the behaviour of $\rho_{\mathrm{S}}$ as $\rho_\mathrm{e}\to0$. The diagonal dashed lines follow $b\rho_\mathrm{e} \rho$. In the first case the function $Q(\rho)-I(\rho_{\mathrm{c}}-\rho)$ initially goes above this line and as $\rho_e\to0$ the maximising argument $\rho_{\mathrm{S}}$ stays away from $0$. In the second case the function stays below the line and $\rho_{\mathrm{S}}\to0$.

Theorems & Definitions (29)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Proposition 2.4
• Definition 2.5
• Corollary 2.6
• Theorem 2.7
• Example 2.9
• Definition 4.1
• Lemma 4.2