Rectangular Matrix Additions in Low and High Temperatures
- 1USA
Abstract
We study the addition of two independent random N× M rectangular matrices with invariant distributions in two limiting regimes, where the parameter β (inverse temperature) tends to infinity and $0$. In the low temperature regime the random singular values of the sum concentrate at deterministic points, while in the high temperature regime, we obtain a law of large numbers for the empirical measures. As a consequence, we obtain a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in the high temperature limit, and degenerate to the classical and free cumulants in special cases.
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BibTeX
@article{xu2026rectangular,
title = {{Rectangular Matrix Additions in Low and High Temperatures}},
author = {Jiaming Xu},
journal = {Symmetry Integrability and Geometry Methods and Applications},
year = {2026},
month = may,
doi = {10.3842/sigma.2026.053},
url = {https://doi.org/10.3842/sigma.2026.053},
note = {CC BY-SA},
}
RIS
TY - JOUR
TI - Rectangular Matrix Additions in Low and High Temperatures
AU - Jiaming Xu
PY - 2026
DA - 2026/05/27
JO - Symmetry Integrability and Geometry Methods and Applications
DO - 10.3842/sigma.2026.053
UR - https://doi.org/10.3842/sigma.2026.053
AB - We study the addition of two independent random N× M rectangular matrices with invariant distributions in two limiting regimes, where the parameter β (inverse temperature) tends to infinity and $0$. In the low temperature regime the random singular values of the sum concentrate at deterministic points, while in the high temperature regime, we obtain a law of large numbers for the empirical measures. As a consequence, we obtain a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in the high temperature limit, and degenerate to the classical and free cumulants in special cases.
SN - 1815-0659
ER -