Fisher-Hartwig asymptotics for non-Hermitian random matrices
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909987325870080 |
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| author | Bourgade, Paul Dubach, Guillaume Hartung, Lisa Keles, Ahmet |
| author_facet | Bourgade, Paul Dubach, Guillaume Hartung, Lisa Keles, Ahmet |
| contents | We prove the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. This formula has applications to random normal matrices with complex spectra: (i) the characteristic polynomial converges to a Gaussian multiplicative chaos random measure on the limiting droplet, in the subcritical phase; (ii) the electric potential converges pointwise to a logarithmically correlated field; (iii) the measure of its level sets (i.e. thick points) is identified; (iv) the associated free energy undergoes a freezing transition.
This establishes emergence of the Liouville quantum gravity measure from free fermions in 2d, and universality with respect to the external potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fisher-Hartwig asymptotics for non-Hermitian random matrices Bourgade, Paul Dubach, Guillaume Hartung, Lisa Keles, Ahmet Probability Mathematical Physics Classical Analysis and ODEs Functional Analysis We prove the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. This formula has applications to random normal matrices with complex spectra: (i) the characteristic polynomial converges to a Gaussian multiplicative chaos random measure on the limiting droplet, in the subcritical phase; (ii) the electric potential converges pointwise to a logarithmically correlated field; (iii) the measure of its level sets (i.e. thick points) is identified; (iv) the associated free energy undergoes a freezing transition. This establishes emergence of the Liouville quantum gravity measure from free fermions in 2d, and universality with respect to the external potential. |
| title | Fisher-Hartwig asymptotics for non-Hermitian random matrices |
| topic | Probability Mathematical Physics Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2512.09123 |