3 research outputs found
Integrable boundaries in AdS/CFT: revisiting the Z=0 giant graviton and D7-brane
We consider the worldsheet boundary scattering and the corresponding boundary
algebras for the Z=0 giant graviton and the Z=0 D7-brane in the AdS/CFT
correspondence. We consider two approaches to the boundary scattering, the
usual one governed by the (generalized) twisted Yangians and the q-deformed
model of these boundaries governed by the quantum affine coideal subalgebras.
We show that the q-deformed approach leads to boundary algebras that are of a
more compact form than the corresponding twisted Yangians, and thus are
favourable to use for explicit calculations. We obtain the q-deformed
reflection matrices for both boundaries which in the q->1 limit specialize to
the ones obtained using twisted Yangians.Comment: 36 pages. v2: minor typos corrected, references updated; v3:
published versio
Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes
We study three instances of log-correlated processes on the interval: the
logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the
Gaussian log-correlated potential in presence of edge charges, and the
Fractional Brownian motion with Hurst index (fBM0). In previous
collaborations we obtained the probability distribution function (PDF) of the
value of the global minimum (equivalently maximum) for the first two processes,
using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the
position of the maximum through its moments. Using replica, this requires
calculating moments of the density of eigenvalues in the -Jacobi
ensemble. Using Jack polynomials we obtain an exact and explicit expression for
both positive and negative integer moments for arbitrary and
positive integer in terms of sums over partitions. For positive moments,
this expression agrees with a very recent independent derivation by Mezzadri
and Reynolds. We check our results against a contour integral formula derived
recently by Borodin and Gorin (presented in the Appendix A from these authors).
The duality necessary for the FDC to work is proved, and on our expressions,
found to correspond to exchange of partitions with their dual. Performing the
limit and to negative Dyson index , we obtain the
moments of and give explicit expressions for the lowest ones. Numerical
checks for the GUE polynomials, performed independently by N. Simm, indicate
encouraging agreement. Some results are also obtained for moments in Laguerre,
Hermite-Gaussian, as well as circular and related ensembles. The correlations
of the position and the value of the field at the minimum are also analyzed.Comment: 64 page, 5 figures, with Appendix A written by Alexei Borodin and
Vadim Gorin; The appendix H in the ArXiv version is absent in the published
versio