{"ID":23681138,"CreatedAt":"2026-09-18T08:49:17.088671394Z","UpdatedAt":"2026-09-18T08:49:17.088671394Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20685","arxiv_id":"2609.20685","title":"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 $\\mathcal R_1$ and $\\mathcal R_2$. 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.","short_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 $\\mathcal R_1$ and $\\mathcal R_2$. At saddle-point leve...","url_abs":"https://arxiv.org/abs/2609.20685","url_pdf":"https://arxiv.org/pdf/2609.20685v1","authors":"[\"Pierre Bousseyroux\",\"Marc Potters\"]","published":"2026-09-17T16:54:38Z","proceeding":"math-ph","tasks":"[\"math-ph\",\"cond-mat.dis-nn\",\"math.PR\"]","methods":"[]","has_code":false}