5 research outputs found
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
Frühdiagenese und Transport in marinen Sedimenten - Bedeutung für den Stofftransfer über die Sediment/Wasser-Grenzfläche
C1 - Journal Articles RefereedThe recent experimental realisation of a one-dimensional Bose gas of ultra
cold alkali atoms has renewed attention on the theoretical properties of the
impenetrable Bose gas. Of primary concern is the ground state occupation of
effective single particle states in the finite system, and thus the tendency
for Bose-Einstein condensation. This requires the computation of the density
matrix. For the impenetrable Bose gas on a circle we evaluate the density
matrix in terms of a particular Painlev\'e VI transcendent in -form,
and furthermore show that the density matrix satisfies a recurrence relation in
the number of particles. For the impenetrable Bose gas in a harmonic trap, and
with Dirichlet or Neumann boundary conditions, we give a determinant form for
the density matrix, a form as an average over the eigenvalues of an ensemble of
random matrices, and in special cases an evaluation in terms of a transcendent
related to Painlev\'e V and VI. We discuss how our results can be used to
compute the ground state occupations