Probabilistic representation and limit theorems for particle numbers of quasi-free states
Abstract
We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose--Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.
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BibTeX
@misc{coudreuse2026probabilistic,
title = {{Probabilistic representation and limit theorems for particle numbers of quasi-free states}},
author = {Fanch Coudreuse and Simone Rademacher and Oliver Tse},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10840},
eprint = {2609.10840},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05747008v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Probabilistic representation and limit theorems for particle numbers of quasi-free states
AU - Fanch Coudreuse
AU - Simone Rademacher
AU - Oliver Tse
PY - 2026
DA - 2026/09/09
PB - HAL
DO - 10.48550/arxiv.2609.10840
UR - https://hal.science/hal-05747008v1
AB - We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose--Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.
ER -