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The Real Ginibre Ensemble with \(k=O(n)\) Real Eigenvalues

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Abstract

We consider the ensemble of real Ginibre matrices conditioned to have positive fraction \(\alpha >0\) of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability \(p^n_{\alpha n}\) that an \(n\times n\) Ginibre matrix has \(k=\alpha n\) real eigenvalues and we characterize the spectral measures of these matrices.

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Notes

  1. For the sake of completeness, we recall that the distance between two probability measures \((\mu ,\nu )\) on \(\mathbbm {C}\) is defined as:

    $$\begin{aligned} d_L(\mu ,\nu )=\inf \left\{ \delta >0\,;\,\mu (A)\le \nu (A^\delta ) \text{ and } \nu (A)\le \mu (A^\delta )\quad \forall A\in \mathcal {B}(\mathbb {C})\right\} \end{aligned}$$

    where \(\mathcal {B}(\mathbb {C}\) is the Borel algebra of \(\mathbbm {C}\) and for \(A\in \mathcal {B}(\mathbbm {C})\), \(A^\delta =\{x;d(x,A)\le \delta \}\).

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Acknowledgments

We thank an anonymous referee for his suggestions on the proof of Theorem 1.

Author information

Authors and Affiliations

  1. Institut Jacques Monod, CNRS UMR 7592, Université Paris Diderot, Paris Cité Sorbonne, 750205, Paris, France

    Luis Carlos García del Molino & Khashayar Pakdaman

  2. Mathematical neuroscience Team, CIRB-Collège de France (CNRS UMR 7241, INSERM U1050, UPMC ED 158, MEMOLIFE PSL), 11 place Marcelin Berthelot, 75005, Paris, France

    Luis Carlos García del Molino & Jonathan Touboul

  3. MYCENAE Team, INRIA Paris-Rocquencourt, 11 place Marcelin Berthelot, 75005, Paris, France

    Jonathan Touboul

  4. Département d’Informatique (DATA), Ecole Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France

    Gilles Wainrib

Authors
  1. Luis Carlos García del Molino
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  2. Khashayar Pakdaman
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Corresponding author

Correspondence to Luis Carlos García del Molino.

Appendices

Appendix

Monte Carlo Algorithm for the Eigenvalues

An efficient method to approximate numerically the minimizer \(\mu _{\alpha }\) and the probability distribution of the proportion of real eigenvalues is to use the Metropolis–Hastings Monte Carlo algorithm. This method consists in constructing an ergodic Markov chain whose stationary distribution is given by (1). Here, we evolve a n-particles system \(z_t\), but in contrast to the log-gas, the dynamics is now discrete, and the transition probability is based on the pdf (1): a new configuration \(z^*\) is drawn by modifying one of the eigenvalues at random and the Markov chain has a transition towards \(z^*\) if \(Q^n(z^*)>Q^n(z_t)\), and otherwise according to a Bernoulli variable of parameter \(\frac{Q^n(z^*)}{Q^n(z_t)}\).

When conditioning on very rare events, (here for instance, a fixed number of real eigenvalues), cases satisfying the constraints have an extremely low probability of being explored, and more refined methods need to be developed in order to access these probabilities. In the present case, the problem is considerably simplified since we dispose of an explicit form of the distribution of the eigenvalues under our constraint. Indeed, the joint probability distribution of Ginibre matrices of size n constrained on having k real eigenvalues \((\lambda _i\;;\; i=1\ldots k)\) (and therefore \(l=(n-k)/2\) pairs of complex eigenvalues \((z_i, i=1\ldots n-k)\)) is given by:

$$\begin{aligned}&\mathbb {P}\big [\lambda _1\ldots \lambda _k, z_1,\ldots ,z_{n-k}\big ]=\tilde{C}_n \prod _{i>j}\vert \lambda _i-\lambda _j\vert \prod _{i>j} \vert z_i-z_j\vert \prod _{i,j}\vert \lambda _i-z_j\vert \\&\qquad \qquad \qquad \times \left( \prod _{i=1}^k\exp (-\lambda _i^2)\prod _{i=1}^{n-k} \exp (-z_i^2)\text {erfc}(\vert z_i-z_i^*\vert /\sqrt{2})\right) ^{1/2}, \end{aligned}$$

where the coefficient \(C_n\) can be found in [14].

Classical Metropolis–Hastings algorithm with Gaussian transitions preserving the nature of the system therefore allow to access directly the distribution of eigenvalues and the probability p(n, k) of the event considered.

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del Molino, L.C.G., Pakdaman, K., Touboul, J. et al. The Real Ginibre Ensemble with \(k=O(n)\) Real Eigenvalues. J Stat Phys 163, 303–323 (2016). https://doi.org/10.1007/s10955-016-1485-0

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