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Upper tail large deviations for extremal eigenvalues of the real, complex and symplectic elliptic Ginibre matrices

Sung-Soo Byun, Yong-Woo Lee, Seungjoon Oh

Abstract

We consider the elliptic Ginibre ensembles in the real, complex and symplectic symmetry classes. As the matrix size tends to infinity, we derive the asymptotic behaviour of the upper tail large deviation probabilities for both the spectral radius and the rightmost eigenvalue. More generally, we obtain asymptotic formulas for the probability that an eigenvalue is found in a prescribed region outside the support of the elliptic law, thereby providing a unified framework in which the results for the spectral radius and the rightmost eigenvalue appear as special cases. The key ingredient of our analysis is the precise asymptotic behaviour of the associated one-point functions, which is of independent interest.

Upper tail large deviations for extremal eigenvalues of the real, complex and symplectic elliptic Ginibre matrices

Abstract

We consider the elliptic Ginibre ensembles in the real, complex and symplectic symmetry classes. As the matrix size tends to infinity, we derive the asymptotic behaviour of the upper tail large deviation probabilities for both the spectral radius and the rightmost eigenvalue. More generally, we obtain asymptotic formulas for the probability that an eigenvalue is found in a prescribed region outside the support of the elliptic law, thereby providing a unified framework in which the results for the spectral radius and the rightmost eigenvalue appear as special cases. The key ingredient of our analysis is the precise asymptotic behaviour of the associated one-point functions, which is of independent interest.
Paper Structure (13 sections, 10 theorems, 108 equations, 7 figures)

This paper contains 13 sections, 10 theorems, 108 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Spectral radius and rightmost eigenvalue
  4. Generalisation to arbitrary domains
  5. Asymptotics of the one-point functions outside the droplet
  6. Exact formulas for finite-N
  7. Planar orthogonal and skew-orthogonal polynomials
  8. Exact identities of the eGinUE correlation functions
  9. Exact identities of the eGinSE correlation functions
  10. Exact identities of the eGinOE correlation functions
  11. Large-N asymptotic analysis
  12. Asymptotics of the one-point functions
  13. Proof of Theorems \ref{['Thm. LDP maxEV']} and \ref{['Thm. LDP maxEV_general domain']}

Key Result

Theorem 2.1

Let $\tau \in [0,1)$ be fixed, and let $N \in \mathbb{N}$. In the eGinOE case, we further assume that $N$ is even. Suppose that $s>1+\tau$. Then we have where Here, $\beta$ is given by def of beta for eGinibres.

Figures (7)

  • Figure 1: Spectra of 50 samples from the elliptic Ginibre ensemble with $N=100$ and $\tau = 0.3$. The solid orange lines indicate the boundary of the ellipse \ref{['def of elliptic law']}.
  • Figure 2: The plot displays the eigenvalues of the eGinOE for $N = 100$ and $\tau = 0.3$. The probability defined in \ref{['def of extremal prob eGinibre']} corresponds to the event that at least one eigenvalue lies in the shaded region, where $s = 1.5$. By \ref{['def of rate Phi_tau']}, for these parameter values we have $\Phi_\tau(s) \approx 0.038$. Hence, by Theorem \ref{['Thm. LDP maxEV']}, the probability of this event is exponentially small, behaving as $e^{-N \Phi_\tau(s)} \approx 0.022$.
  • Figure 3: The plots show the graphs of $s \mapsto \Phi_\tau(s)$ for $s > 1+\tau$. The black dotted lines indicate the point $s = 1+\tau$.
  • Figure 4: Surface and contour plot of $z \mapsto \Omega(z)$ in $E^c$, where $\tau = 0.3$.
  • Figure 5: Plots of the normalised one-point function $R_N^\mathbb{C}(z) / (\sqrt{N} e^{-N \Omega(z)})$ for the eGinUE with $\tau=0.3$, compared with the analytic result from Proposition \ref{['Prop_eGinUE asymp']}. The comparison is shown through cross-sections with either $x$ or $y$ fixed.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Theorem 2.1: Upper tail large deviation probabilities for spectral radius and rightmost eigenvalue
  • Remark 1: Integral representation
  • Remark 2: Finite-$N$ corrections
  • Theorem 2.2: Upper tail large deviation probabilities in a general domain
  • Remark 3: Rate function as a solution to an obstacle problem
  • Remark 4: Generalisation to random normal matrix models
  • Lemma 3.1: cf. Proposition 2.3 in LR16
  • Lemma 3.2: cf. Proposition 1.1 and Lemma 3.2 in BE23
  • Lemma 3.3: cf. Eq. (5.18) in AP14
  • Lemma 4.1: cf. Lemma 2.5 in LR16
  • ...and 9 more