Phase transitions in non-Hermitian spherical integrals
Abstract
We study the large-N asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms ℛ1 and ℛ2. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.
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BibTeX
@misc{bousseyroux2026phase,
title = {{Phase transitions in non-Hermitian spherical integrals}},
author = {Pierre Bousseyroux and Marc Potters},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20685},
eprint = {2609.20685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20685},
}
RIS
TY - UNPB
TI - Phase transitions in non-Hermitian spherical integrals
AU - Pierre Bousseyroux
AU - Marc Potters
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20685
UR - https://arxiv.org/abs/2609.20685
AB - We study the large-N asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms ℛ_1 and ℛ_2. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.
ER -