Abstract. We report on the observation of three high-altitude cusp crossings by the Cluster spacecraft under steady northward IMF conditions. The focus of this study is on the exterior cusp and its boundaries. At the poleward edge of the cusp, large downward jets are present; they are characterized by a dawn-dusk component of the convection velocity opposite to the IMF B y direction and a gradual evolution (velocity filter effect) corresponding to an injection site located at the high-latitude magnetopause tailward of the cusp, with subsequent sunward convection. As one moves from the poleward edge into the exterior cusp proper, the plasma gradually becomes stagnant as the result of the mirroring and scattering of the aforementioned plasma flows. The existence of such a stagnant region (Stagnant Exterior Cusp: SEC) is found in all events studied here even when the IMF B y is large and the clock angle is ∼90 • . The SEC-magnetosheath boundary appears as a spatial structure that has a normal component of the magnetic field pointing inward, in accordance with a probable connection between the region and the magnetosheath (with northward field). This boundary generally has a deHoffmann-Teller velocity that is slow and oriented sunward and downward, compatible with a discontinuity propagating from a location near the high-latitude magnetopause. Although the tangential stress balance is not always satisfied, the SEC-magnetosheath boundary is possibly a rotational discontinuity. Just outside this boundary, there exists a clear sub-Alfvénic plasma depletion layer (PDL). These results are all consistent with the existence of a nearly steady reconnection site at the high-latitude magnetopause tailward of the cusp. We suggest that the stability of the external discontinuity (and of the whole region) is maintained by the presence of the sub-Alfvénic PDL. However, examinationCorrespondence to: B. Lavraud (lavraud@lanl.gov) of the electron data shows the presence of heated electrons propagating parallel to the magnetic field (upward) just outside of the SEC-magnetosheath boundary. This appears inconsistent with their source being the northern lobe reconnection site. Finally, the definition of the magnetopause at high latitudes is revisited. To define the SEC-magnetosheath boundary as the magnetopause would lead to the misnaming of the "exterior cusp".

Abstract. Recently much effort has been made towards the introduction of nonHermitian random matrix models respecting P T -symmetry. Here we show that there is a one-to-one correspondence between complex P T -symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. We conjecture that these ensembles represent universality classes for P Tsymmetric matrices. For the case of 2 × 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues.In recent years there has been a surge of research interest in P T -symmetric quantum theories, accompanied by a multitude of experimental applications and realisations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For finite-dimensional systems represented by matrices, P Tsymmetry is equivalent to the reality of the characteristic polynomial [18]. That is, P T -symmetric matrices have either real or complex conjugate eigenvalues. Their eigenvectors are orthogonal with respect to a suitably defined CP T inner product [19]. It has recently been conjectured that P T -symmetry is closely related to split-quaternionic extensions of quantum theory [20,21]. Here we show that splitquaternionic extensions of Hermitian matrices are indeed a natural representation of P T -symmetric matrices. This equivalence allows us to introduce new P T -symmetric random matrix ensembles.In conventional quantum systems, random matrices play an important role due to their ability to describe spectral fluctuations in sufficiently complicated systems [22]. In particular, the famous Bohigas-Giannoni-Schmit conjecture states that the spectral fluctuations of quantum systems with chaotic classical counterparts are similar to those of certain random matrices [23,24]. There are three important universality classes for Hermitian quantum systems, depending on the time-reversal properties of the system, corresponding to the Gaussian orthogonal, unitary, and symplectic ensembles [25]. Non-Hermitian random matrix models, on the other hand, whose eigenvalues are in general complex, are widely studied, and have applications ranging from dissipative quantum systems and scattering theory to quantum chromodynamics (see, e.g., [26] and references therein). Several attempts towards defining P Tsymmetric random matrices and identifying universality classes for P T -symmetric systems have been made [27][28][29][30][31][32]. Most of them are restricted to 2 × 2 matrices, due to the lack of a natural parameterisation of larger P T -symmetric matrices. Here we introduce the split-complex and split-quaternionic versions of the Gaussian unitary arXiv:1505.07810v2 [math-ph]

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