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Maximum of the Characteristic Polynomial of the Ginibre Ensemble

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Abstract

We compute the leading asymptotics for the maximum of the (centered) logarithm of the absolute value of the characteristic polynomial, denoted \(\Psi _N\), of the Ginibre ensemble as the dimension of the random matrix N tends to \(\infty \). The method relies on the log-correlated structure of the field \(\Psi _N\) and we obtain the lower-bound for the maximum by constructing a family of Gaussian multiplicative chaos measures associated with certain regularization of \(\Psi _N\) at small mesoscopic scales. We also obtain the leading asymptotics for the dimensions of the sets of thick points and verify that they are consistent with the predictions coming from the Gaussian Free Field. A key technical input is the approach from Ameur et al. (Ann Probab 43(3):1157–1201, 2015) to derive the necessary asymptotics, as well as the results from Webb and Wong (Proc Lond Math Soc (3) 118(5):1017–1056, 2019).

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Notes

  1. We briefly review the definition of the GFF in Sect. 2.1.

  2. A random \(N\times N\) matrix sampled from the Haar measure on the unitary group.

  3. A random \(N\times N\) Hermitian matrix with independent Gaussian entries suitably normalized.

  4. They obtain tightness of the appropriately centered maximum for both real and imaginary part of the logarithm of the characteristic polynomial. See also [42] for the asymptotics of the measures of thick points by a different approach.

  5. See Theorem 1.4 below.

  6. The concept of thick points is crucial to describe the geometric properties of log-correlated fields. Informally, these points corresponds to the extremal values of the field.

  7. That is in the subcritical phase — the critical value being \(\gamma _*=\sqrt{8}\) as in (1.15) or in Theorem 2.2 below.

  8. This means that \(\phi (x)\) is a smooth probability density function which only depends on |x| with compact support in the disk \(\mathbb {D}_{\epsilon _0}\). Note that we can work with any such mollifier.

  9. This follows from the fact that since the mollifier \(\phi \) is radial and compactly supported, \(\psi _\epsilon (z) = \log |z|\) for all \(|z| \ge \epsilon \) and for any \(\epsilon >0\).

  10. We chose this unusual normalization in order to match with formula (1.10).

  11. This formula for the limiting covariance in the Rider–Viràg CLT holds for test functions which are harmonic outside of \(\mathbb {D}\) [53]. In particular, it can be applied to (2.1) if \(z\in \mathbb {D}\) and \(\epsilon >0\) is small enough. Then, one deduces the counterpart of (2.7)–(2.8) holds for the field (2.3) which is supported in \(\mathbb {D}_{2\epsilon _0}\) by linearity.

  12. Note that our normalization does not match with the standard convention for log-correlated fields used in [22, Section 3]. Actually, we apply [22, Theorem 3.4] to the field \(\mathrm {X}(z)= \sqrt{2} \mathrm {X}(g_N^z)\) — this explains why the critical value is \(\gamma ^*= \sqrt{8}\) as well as the factor \(\frac{\gamma ^*}{2}\) in (2.11).

  13. Note that our approximations are more precise than in [4].

  14. Note that we have \(\Delta \left( \log \sqrt{1+|x|^2} \right) = \frac{4}{(1+|x|^2)^2} >0\) for \(x\in \mathbb {C}\).

  15. Here we used that \(\chi _z=1\) on \(\mathbb {D}(z;\delta )\) so that \(\overline{\partial }f=0\) and that \(\overline{\partial }f = \overline{\partial }U\) since \(U-f \in \mathscr {P}_N\) by definition of U.

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Acknowledgements

G.L. is supported by the University of Zurich Forschungskredit Grant FK-17-112 and by the Grant SNSF Ambizione S-71114-05-01. G.L. wishes to thank P. Bourgade for insightful discussions about the problem considered here and the referee for interesting comments which helped to improve the presentation of this article and for pointing out several references.

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    Gaultier Lambert

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Lambert, G. Maximum of the Characteristic Polynomial of the Ginibre Ensemble. Commun. Math. Phys. 378, 943–985 (2020). https://doi.org/10.1007/s00220-020-03813-1

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