Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE
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arXiv
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| Natura: | Preprint |
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2024
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| author | Byun, Sung-Soo Seo, Seong-Mi Yang, Meng |
| author_facet | Byun, Sung-Soo Seo, Seong-Mi Yang, Meng |
| contents | We consider a planar Coulomb gas ensemble of size $N$ with the inverse temperature $β=2$ and external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18983 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE Byun, Sung-Soo Seo, Seong-Mi Yang, Meng Mathematical Physics Complex Variables Probability We consider a planar Coulomb gas ensemble of size $N$ with the inverse temperature $β=2$ and external potential $Q(z)=|z|^2-2c \log|z-a|$, where $c>0$ and $a \in \mathbb{C}$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1) N \times (c+1) N$ conditioned to have deterministic eigenvalue $a$ with multiplicity $cN$. Depending on the values of $c$ and $a$, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-$N$ expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform. |
| title | Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE |
| topic | Mathematical Physics Complex Variables Probability |
| url | https://arxiv.org/abs/2402.18983 |