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In this paper, we rigorously investigate the reduced density matrix (RDM)associated to the ideal Bose gas in harmonic traps. We present a method basedon a sum-decomposition of the RDM allowing to treat not only the isotropictrap, but also general anisotropic traps. When focusing on the isotropic trap,the method is analogous to the loop-gas approach developed by W.J. Mullin in[38]. Turning to the case of anisotropic traps, we examine the RDM for someanisotropic trap models corresponding to some quasi-1D and quasi-2D regimes.For such models, we bring out an additional contribution in the local densityof particles which arises from the mesoscopic loops. The close connection withthe occurrence of generalized-BEC is discussed. Our loop-gas-like approachprovides relevant information which can help guide numerical investigations onhighly anisotropic systems based on the Path Integral Monte Carlo (PIMC)method.

Rigorous investigation of the reduced density matrix for the ideal Bose gas in harmonic traps by a loop-gas-like approach