Abstract:In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.
From: Pierre Bousseyroux [view email]
[v1]
Wed, 14 Jan 2026 10:40:27 UTC (37 KB)
[v2]
Thu, 14 May 2026 10:32:13 UTC (38 KB)
[v3]
Tue, 1 Sep 2026 13:09:32 UTC (39 KB)
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