An Explicit Large Deviation Analysis of the Spatial Cycle Huang–Yang–Luttinger Model

Abstract

We introduce a family of ‘spatial’ random cycle Huang–Yang–Luttinger (HYL)-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. Here, spatial refers to the Poisson reference process of random cycle weights. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose–Einstein condensation. The main focus is a large deviation analysis for the diverging counter term where we identify three different regimes depending on the scale of divergence with respect to the main large deviation scale. Our analysis derives explicit bounds in critical regimes using the Poisson nature of the random cycle distributions.

Formation of infinite loops for an interacting bosonic loop soup

“…The study of the grand-canonical ensemble for the partial loop HYL model in [AD21] derived a large deviation principle -a lower resolution result than that derived here. Nevertheless, it was sufficient to derive the 'condensate' density of that loop soup model.…”

“…Not only does it replace momentum eigenstate occupation numbers with loop type occupation numbers, but the second term omits more and more lengths of loop as the thermodynamic limit is taken. Nevertheless, it has been proven in [AD21] that the thermodynamic pressure of this loop model is precisely equal to the thermodynamic pressure of the HYL Bose model derived in [vdBDLP90].…”

“…We first state the theorem before giving the conditions required for G. Let I be the rate-function for the particle number with no interaction, given in Equation (4.11). Note that this can be also written as the sum of the free energy and the pressure of the system (from a simpler application of the techniques in [AD21]).…”

“…That work considered the case of the free Bose gas as well as mean-field case, where the interaction between loops is given by the square of the total particle number. In this work, we consider the (partial) HYL energy, a non mean-field interaction between loops, inspired by the works of [HYL57,Lew86,AD21], mimicking the repulsion between particles. Such Bose soup behaves qualitatively different to free and mean-field cases.…”

“…In contrast to [Vog21], we chose R d instead of Z d as the underlying space, to demonstrate the robustness of the proofs and to align with [AFY19,AD21]. For measurable Λ ⊂ R d , the measure governing the distribution of individual loops is called the bosonic loop measure, given by…”