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We study three instances of log-correlated processes on the interval: thelogarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, theGaussian log-correlated potential in presence of edge charges, and theFractional Brownian motion with Hurst index $H \to 0$ (fBM0). In previouscollaborations we obtained the probability distribution function (PDF) of thevalue of the global minimum (equivalently maximum) for the first two processes,using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of theposition of the maximum $x_m$ through its moments. Using replica, this requirescalculating moments of the density of eigenvalues in the $\beta$-Jacobiensemble. Using Jack polynomials we obtain an exact and explicit expression forboth positive and negative integer moments for arbitrary $\beta >0$ andpositive integer $n$ in terms of sums over partitions. For positive moments,this expression agrees with a very recent independent derivation by Mezzadriand Reynolds. We check our results against a contour integral formula derivedrecently 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 thelimit $n \to 0$ and to negative Dyson index $\beta \to -2$, we obtain themoments of $x_m$ and give explicit expressions for the lowest ones. Numericalchecks for the GUE polynomials, performed independently by N. Simm, indicateencouraging agreement. Some results are also obtained for moments in Laguerre,Hermite-Gaussian, as well as circular and related ensembles. The correlationsof the position and the value of the field at the minimum are also analyzed.

Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes