Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

Abstract

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 H → 0 (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 freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum x m through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β > 0 and positive integer n 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 1 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 n → 0 and to negative Dyson index β → −2, we obtain the moments of x m 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.With Appendix 1 written

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“…We will return to the meaning of Z b later in E. When approaching the critical temperature b → 1 − , it is now known [39] that the wellbehaved object is not the partition function Z b but its derivative (∂ b Z b )| b=1 . This echoes the fact that, in all known exact solved cases [5,[30][31][32][33][34], the analytically continued continuum integral Z n b has a zero of order n at b = β c = 1. Therefore we define…”

“…In spite of the product form, we do not have an obvious interpretation of (24) in terms of statistical independence; in particular, the minimum and the first gap V min,1 − V min are not uncorrelated, see (32) below. Nevertheless, the two terms of the RHS have contrasted nature, as we detail below:…”

“…The treatment was based on assuming the validity of the freezing scenario conjecturing that in the thermodynamic limit the whole distribution of the Gumbelconvoluted free-energy F − G/β, with F being the free energy at temperature 1/β, and G being a standard Gumbel variable independent of F , is freezing, that is becomes temperature-independent below the transition temperature, modulo a translation. Further work [30] observed the co-existence of freezing with the duality-invariance property, and extended the above freezing scenario to the freezing-duality conjecture (FDC), which has been checked numerically for several observables in various logREMs [30][31][32][33][34]. Although a relation between FDC and 1RSB was hinted briefly in [31], up to now very little has been worked out explicitly in this direction.…”

“…An example of less trivial l(ξ) and c(ξ) is the interval model [30,32]. Its general discrete definition depends on four parameters c 0 , c 1 ,…”

“…FDC promotes this idea further by boldly suggesting that any duality invariant thermodynamic observable must freeze beyond the self-dual point β = 1. It has been successfully applied to calculate free energy distribution, moments of minimum position distribution [31,32], and more involved quantities [34].…”