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Spiking the random matrix hard edge

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Abstract

We characterize the limiting smallest eigenvalue distributions (or hard edge laws) for sample covariance type matrices drawn from a spiked population. In the case of a single spike, the results are valid in the context of the general \(\beta \) ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion (\(\beta =1,2,\) or 4) ensembles. The limit laws are described in terms of random integral operators, and partial differential equations satisfied by the corresponding distribution functions are derived as corollaries. We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy–Widom laws). The latter were first described at \(\beta =2\) by Baik, Ben Arous, and Peché (Ann Probab 33:1643–1697, 2005), and from a unified \(\beta \) random operator point of view by Bloemendal and Virág (Probab Theory Relat Fields 156:795–825, 2013; Ann Probab arXiv:1109.3704, 2011).

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Notes

  1. In [19] the generator is denoted by \(-{\mathfrak {G}}\), with \({\mathfrak {G}}^{-1}\) reserved for the Green’s operator. This should’t cause confusion here.

  2. In [19] we also flipped the indexing by conjugating as in \(B \mapsto S B S^{-1}\) with \(S_{ij} = (-1)^i \delta _{i+j - n-1}\). With the benefit of hindsight it is more convenient not to make this extra move.

  3. Throughout we use the convention that exponents of the form \(b \mapsto b^{\alpha {+}}\) or \(b^{\alpha -}\) indicate that the appraisal in question holds for \(\alpha \pm \epsilon \) with any choice of \(\epsilon > 0\). In the present example, [19] reports an exponent of 3 / 4 in the bounding function \(\phi \), but the proof there shows that this can be replaced by anything greater than 1 / 2.

  4. Throughout \([\cdot ]\) denotes the (appropriate) integer part.

  5. This computation is more or less reproduced in [22]. It should be clear afterwards how everything goes through in the quaternion case (in the real case there are no subtleties—the \(\hbox {diag}(Q_x)\) term is not present and the Brownian motion \(B_x\) is isotropic).

  6. Here we are tacitly assuming that the limiting starting point w is finite, if \(a^{-2/3}c - a^{1/3} \rightarrow \infty \) an additional approximation step is needed, but this has been carried out in [19].

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Acknowledgments

Many thanks to Tom Kurtz, Benedek Valkó, and Ofer Zeitouni for helpful discussions. Thanks as well to the anonymous referee for a careful and thoughtful review. B.R. was supported in part by NSF Grant DMS-1340489 and Grant 229249 from the Simons Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Universidad de Costa Rica, San José, Costa Rica

    José A. Ramírez

  2. Department of Mathematics, Temple University, Philadelphia, USA

    Brian Rider

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  1. José A. Ramírez
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Ramírez, J.A., Rider, B. Spiking the random matrix hard edge. Probab. Theory Relat. Fields 169, 425–467 (2017). https://doi.org/10.1007/s00440-016-0733-1

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  • DOI: https://doi.org/10.1007/s00440-016-0733-1

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