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On the distribution of the largest real Eigenvalue for the real Ginibre Ensemble
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Poplavskyi, Mihail, Tribe, Roger and Zaboronski, Oleg V. (2017) On the distribution of the largest real Eigenvalue for the real Ginibre Ensemble. Annals of Applied Probability, 27 (3). pp. 1395-1413. doi:10.1214/16-AAP1233 ISSN 1050-5164.
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Official URL: http://doi.org/10.1214/16-AAP1233
Abstract
Let N−−√+λmaxN+λmax be the largest real eigenvalue of a random N×NN×N matrix with independent N(0,1)N(0,1) entries (the “real Ginibre matrix”). We study the large deviations behaviour of the limiting N→∞N→∞ distribution P[λmax<t]P[λmax<t] of the shifted maximal real eigenvalue λmaxλmax. In particular, we prove that the right tail of this distribution is Gaussian: for t>0t>0, P[λmax<t]=1−14erfc(t)+O(e−2t2).
P[λmax<t]=1−14erfc(t)+O(e−2t2).
This is a rigorous confirmation of the corresponding result of [Phys. Rev. Lett. 99 (2007) 050603]. We also prove that the left tail is exponential, with correct asymptotics up to O(1)O(1): for t<0t<0,
P[λmax<t]=e122π√ζ(32)t+O(1),
P[λmax<t]=e122πζ(32)t+O(1),
where ζζ is the Riemann zeta-function.
Our results have implications for interacting particle systems. The edge scaling limit of the law of real eigenvalues for the real Ginibre ensemble is a rescaling of a fixed time distribution of annihilating Brownian motions (ABMs) with the step initial condition; see [Garrod, Poplavskyi, Tribe and Zaboronski (2015)]. Therefore, the tail behaviour of the distribution of X(max)sXs(max)—the position of the rightmost annihilating particle at fixed time s>0s>0—can be read off from the corresponding answers for λmaxλmax using X(max)s=D4s−−√λmaxXs(max)=D4sλmax.
| Item Type: | Journal Article | ||||||
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Journal or Publication Title: | Annals of Applied Probability | ||||||
| Publisher: | Institute of Mathematical Statistics | ||||||
| ISSN: | 1050-5164 | ||||||
| Official Date: | 19 July 2017 | ||||||
| Dates: |
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| Volume: | 27 | ||||||
| Number: | 3 | ||||||
| Page Range: | pp. 1395-1413 | ||||||
| DOI: | 10.1214/16-AAP1233 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 5 October 2016 | ||||||
| Date of first compliant Open Access: | 27 November 2017 | ||||||
| Related URLs: | |||||||
| Open Access Version: |
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