Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 053, 51 pages      arXiv:2303.13812      https://doi.org/10.3842/SIGMA.2026.053

Rectangular Matrix Additions in Low and High Temperatures

Jiaming Xu
Columbus, USA

Received April 09, 2025, in final form May 06, 2026; Published online May 27, 2026

Abstract
We study the addition of two independent random $N\times M$ rectangular matrices with invariant distributions in two limiting regimes, where the parameter $\beta$ (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.

Key words: rectangular random matrices; $\beta$-ensemble; symmetric functions; Dunkl operators.

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