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- W4308169164 abstract "In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $exp[sqrt{nm}theta X_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $theta$ is a real number, $M_n$ is a certain $ntimes m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matrices with respective sizes $ntimes n$, $mtimes m$. We prove that when the singular law of $M_n$ converges to a probability measure $mu$, for $theta$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $mu$. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms." @default.
- W4308169164 created "2022-11-08" @default.
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- W4308169164 date "2009-09-01" @default.
- W4308169164 modified "2023-09-26" @default.
- W4308169164 title "Rectangular R-transform as the limit of rectangular spherical integrals" @default.
- W4308169164 doi "https://doi.org/10.48550/arxiv.0909.0178" @default.
- W4308169164 hasPublicationYear "2009" @default.
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