key: cord-0188579-vbn77ayy authors: Fyodorov, Yan V.; Doussal, Pierre Le title: Statistics of extremes in eigenvalue-counting staircases date: 2020-01-13 journal: nan DOI: nan sha: e5e5326b5507a657081f873d323691c44d65381f doc_id: 188579 cord_uid: vbn77ayy We consider the number ${cal N}(theta)$ of eigenvalues $e^{i theta_j}$ of a random unitary matrix, drawn from CUE$_{beta}(N)$, in the interval $theta_j in [theta_A,theta]$. The deviations from its mean, ${cal N}(theta) - mathbb{E}({cal N}(theta))$, form a random process as function of $theta$. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture, supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any $beta>0$. It exhibits combined features of standard counting statistics of fermions (free for $beta=2$ and with Sutherland-type interaction for $betane 2$) in an interval and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. The $beta=2$ results are expected to apply to the statistics of zeroes of the Riemann Zeta function. We consider the number N (θ) of eigenvalues e iθ j of a random unitary matrix, drawn from CUE β (N ), in the interval θj ∈ [θA, θ]. The deviations from its mean, N (θ) − E(N (θ)), form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any β > 0. It exhibits combined features of standard counting statistics of fermions (free for β = 2 and with Sutherland-type interaction for β = 2) in an interval and extremal statistics of the fractional Brownian motion with Hurst index H = 0. The β = 2 results are expected to apply to the statistics of zeroes of the Riemann Zeta function. Characterizing the full counting statistics of the fluctuations of the number N of 1d fermions in an interval is important in numerous physical contexts, both for ground state and dynamical properties. It appears e.g. in shot noise [1] , in fermion chains [2, 3] , in interacting Bose gases [4] , in non-equilibrium Luttinger liquids [5] , in trapped fermions [6] [7] [8] , and for studying related observables, such as the entanglement entropy [9] [10] [11] or the statistics of local magnetization in quantum spin chains [12] . An equivalent problem can be formulated as counting eigenvalues of large random matrices. As is well known since Dyson's work [13] , such eigenvalues behave as classical particles with 1-d Coulomb repulsion at inverse temperature β > 0. Namely, consider a unitary N × N matrix U and denote the corresponding unimodular eigenvalues as z j = e iθj , j = 1, . . . , N , with phases θ i ∈] − π, π]. Then for any given β > 0 one can construct the so-called Circular Unitary β-Ensemble CUE β (N ) in such a way that the expectation of a function depending only on the eigenvalues of U will be given by where F ≡ F (θ 1 , . . . , θ n ). For β = 2 such matrices can be thought of as drawn uniformly according to the corresponding Haar's measure on U (N ), whereas for a generic β > 0 the explicit construction is more involved, see [14] . For any β > 1, the r.h.s of (1) equals the quantum expectation value of F in the ground state of N spinless fermions, of coordinates θ i on the unit circle, described by the Sutherland Hamiltonian [15] , 8 sin 2 θ i −θ j 2 . For β = 2, Eq (1) thus describes non-interacting fermions, while for β = 2 the fermions interact, via an inverse square distance pairwise potential. Let us now define the number of eigenvalues/fermions, N θ A (θ), in the interval θ j ∈ [θ A , θ] as (2) As a function of θ this is a staircase with unit jumps upwards at random positions θ j ∈ [θ A , θ]. The mean slope (i.e. the mean density of eigenvalues/fermions) being constant, the mean profile is E(N θ A (θ)) = N (θ−θ A ) 2π . In a given random matrix realization/sample one can define the deviation to the mean, δN θ A (θ) = N θ A (θ) − E(N θ A (θ)), and study it as a random process as a function of θ, i.e. as a function of the length of the interval θ − θ A . From the view of such a process, the standard results on fermion counting statistics [2] , encoding the full distribution of δN θ A (θ) for a fixed value of θ, is a very local information. Such information is clearly insufficient for understanding various non-local properties of the process, such as characterizing maximal deviation of the staircase from its mean, i.e. max θ∈ After appropriate normalization this is simply the Kolmogorov-Smirnov (KS) statistics, an important outstanding open problem for spectra of random matrices (see recent discussion and references in [16] ). The interest in KS-like statistics provides a strong motivation to study separately the value distribution for the maximum & minimum for the process δN θ A (θ). In this Letter, we provide a solution for such a problem, by explicitly calculating the cumulants of the probability density function (PDF) for the maximum value defined as on an interval [θ A , θ B ] ⊂] − π, π], of a fixed length = θ B − θ A . To derive the PDF of δN m in the limit N 1 we will show that for scales larger than 1/N the process δN θ A (θ) is very close to a special version of 1D log-correlated Gaussian field, the so called fractional Brownian Motion with Hurst index H = 0 which was defined in [17] and that we denote as fBm0. The properties of the extrema of log-correlated processes of such type have been under investigation recently [18, 19] and we will capitalize on this knowledge. However it turns out that the relation to fBm0 alone is insufficient to fully determine the statistics of the maximum for the difference between the counting staircase and its mean. Namely, we will demonstrate that although the process δN θ A (θ) for large N 1 is very close to the fBm0 at different points, the non-Gaussian features which characterize its single-point statistics show up in a non-trivial way in the PDF of its maximum δN m . These single-point features are inherited from the discrete nature of the number of fermions/eigenvalues as exemplified e.g. in fermion counting statistics [2] . Another interesting question is to determine the PDF of the location of the maximum in (3), denoted θ m ∈ [θ A , θ B ]. We provide some results below. We now describe our main findings by first assuming that the Dyson parameter is rational and can be represented as β/2 = s/r where s and r are mutually prime, and relaxing this assumption later on. We find that, for any fixed interval, the mean value of the maximum δN m defined in (3) exhibits, for N → ∞, the universal behavior of the log-correlated fields [20] [21] [22] [23] : where c (β) = O(1) is an unknown -dependent constant. The variance for the maximum δN m exhibits to the leading order the extensive universal logarithmic growth typical for pinned log-correlated fields [18] , on top of which we can evaluate the corrections of the order of unity: Finally, the higher cumulants converge to a finite limit as N → ∞: where the constants C k ( ) = O(1) depend on the length of the interval and will be given below in two limiting cases. The −independent constantsC (β) k for k ≥ 2 are given byC where Here G(z) denotes the standard Barnes function satisfying G(z + 1) = Γ(z)G(z), with G(1) = 1. Note that all the odd coefficientsC (β) 2k+1 vanish. Specifying for β = 2, one has A 2 (t) = G(1+it), leading toC (3) . Notably, using (7), (8), we were able to obtain a formula for theC (β) k as single infinite series [24] , which shows that they are smooth as a function of the Dyson parameter β, thus relaxing the assumption of rationality. As discussed below, the factors A β (t), hencẽ C (β) k , are intimately but non-trivially related to the cumulants of the number of fermions (free for β = 2 and with Sutherland-type interaction for β = 2) in a mesoscopic interval of the circle. By contrast the factors C k ( ) are β-independent and originate from the problem of the maximum of a fBm0 on the interval [θ A , θ B ]. For the -dependent constants we obtain explicit formula in two cases: (i) maximum over the full circle = 2π. In that case [θ A , θ B ] =] − π, π] and we find for any k ≥ 2 which is related to the fBm0 bridge on ] − π, π] studied in [18] (ii) maximum over a mesoscopic interval 1 N 1. For k ≥ 2 we obtain in this regime Note that the variance depends logarithmically on at small , whereas higher cumulants have limits as → 0. This result is related to the fBm0 on an interval, with one pinned and one free end, studied in [18] . Finally, addressing the question of the location of the maximum in (3), θ m ∈ [θ A , θ B ], let us define y m = (θ m − θ A )/ . We predict the PDF of y m to be symmetric around 1 2 , with E(y 2 m ) = 17 50 and E(y 4 m ) = 311 1470 , thus deviating from the uniform distribution. For the full circle we do find a uniform distribution for θ m [25] . However, joint moments for the position and value of the maximum, obtained in [24], do show the effect of pinning at θ = θ A . It is natural to conjecture that many of our results for β = 2 should equally describe the statistics of the counting staircases for the nontrivial zeroes t n of the Riemann zeta-function ζ(1/2 + it) in intervals of the critical line t ∈ R. Such zeroes are known to be extremely faithful to the random matrix statistics when analyzed in appropriate scales [26] . This observation underlies a fruitful line of applications of RMT to understand ensuing features of ζ(1/2 + it). Along this way, Keating and Snaith [27] convincingly demonstrated that statistical properties of the value distribution of the Riemann zeta function on the critical line can be very successfully modelled by invoking similar properties of the random characteristic polynomials for CUE β=2 (N ). In this context, revealing and exploiting nontrivial relations with the log-correlated fields turned out to be a very fruitful line of research , see e.g. [28] for a review. Indeed, the logarithm of ζ(1/2 + it), t ∈ R has been shown to behave statistically like a log-correlated field over t in mesoscopic intervals containing an appropriate number of nontrivial zeroes [29, 30] , sharing this property with the logarithm of the characteristic polynomial defined in Eq.(13) [31, 32] . To provide a proper background for our calculation, it is worth starting with elucidating the relation to fBm0. To this end, let us recall that the process δN θ A (θ) is exactly given by the difference where ξ N (θ) is the characteristic polynomial defined as As has been shown in [31] for β = 2 (see [32] for general β > 0) the joint probability density of Im log ξ N (θ) at two fixed distinct points θ 1 = θ 2 converges as N → +∞ to that of a Gaussian process W β (θ) of zero mean and covariance which is a particular instance of the 1D log-correlated Gaussian field. Since (12) implies that δN θ A (θ = θ A ) = 0 in any realization, the relevant object is the so-called pinned log-correlated process closely related to fBm0. We shall see however [24] that naively replacing the difference δN θ A (θ) with its Gaussian approxima- (closely related to the so-called bosonisation of the fermionic problem) is not sufficient for the purpose of characterizing the maximum of the process. Gaussian fields and processes characterized by a logarithmic covariance structure appear in many contexts, to name a few: chaos and turbulence [33] , the statistical mechanics of branching random walks and polymers on trees [20, 21] , disordered systems with multifractal structure [34, 35] , probabilistic descriptions of two-dimensional gravity [36, 37] . Here, we will build on the progress made in the characterization of the extrema of log-correlated fields. Early works revealed the problem to be intimately connected to a remarkable freezing transition [20, 21, 34] . Through exact solutions, it led to predictions for the PDF of the maximum value of a log-correlated field on the circle and on the interval [38, 39] , involving the so-called freezing duality conjecture (FDC) (see [19] for an extensive discussion of this conjecture). This led to intensive studies and further results in theoretical and mathematical physics [22, [40] [41] [42] [43] [44] and probability [23, [45] [46] [47] [48] [49] [50] [51] [52] [53] . One should remember that log-correlated fields are essentially random generalised functions (distributions) and when discussing their value distribution and extrema one necessarily works with reg-ularizations of such highly singular object. In particular, the log of a random characteristic polynomial for a large but finite N provides such a natural regularization [32, 54, 55] augmenting the covariance (14) with the variance E(W (θ) 2 ) = β −1 log N . Together with the aforementioned applications to the statistics of the Riemann zeta function on the critical line, the new perspective has recently attracted a lot of attention, with emphasis on statistics of extremes of the associated objects [32, [47] [48] [49] [56] [57] [58] [59] [60] [61] [62] [63] . Let us however stress, that none of these studies yet addressed the extremes associated with the eigenvalue/zeros counting function. To study the maximum of the random field δN (θ) we follow the ideas of [19, 38, 39, 56, 57] and introduce the following partition sum: thus mapping the search of the maximum to a statistical mechanics problem. The " inverse temperature" associated with such partition sum is equal to −2πb β/2, and we choose b > 0 since we are studying here the maximum retrieved from the free energy F for b → +∞ as To study the statistics of the associated free energy we start with considering the integer moments of Z b given by The expectation value in (17) over the CUE β (N ) computed using (1) has the form E[ This can be further rewritten for any φ a , θ, θ A ∈] − π, π] with φ a > θ A as where we define the arg function as This form is exactly the one which allows to extract the leading asymptotic of E[ N j=1 g(θ j )] in large N limit using the extended version [64] of the Fisher-Hartwig conjecture [65] , the case β = 2 [66] proved rigorously in [67] . Specifying the expressions in [64] to our case gives for N → +∞ and nb 2 < 1 where we have assumed that the integral is dominated by all distinct φ a (which can be checked to be the case for b < 1) and the function A β (b) is defined in (8) . It is important to note that had we used instead in our calculation an approximation replacing the difference δN θ A (θ) in the large−N limit with the logarithmically correlated Gaussian process W β (θ) defined via (12) -(14), we would reproduce the Coulomb gas factors in (21) but completely miss the product of factors A β (b), see [24] . Hence, this product encapsulates the residual non-Gaussian features of the associated process. Let us first discuss the simplest case n = 1 and β = 2 implying s = r = 1. In this case the integrand in (21) is This formula encodes the one-point distribution of the field N θ A (θ) [68] and can be interpreted as the generating function for the full counting statistics for the number of free fermions in an interval of size θ−θ A . For small θ−θ A this coincides with the result in [2] in the low density limit k F → 0 (setting 2Lk F = N ). Similarly, for n = 1 and a general rational β/2 we get the full counting statistics for the number of fermions in the Sutherland model, which seems not to be addressed in the literature apart from β = 4 [69] . Further progress is possible in the two cases when the Coulomb integrals in (21) can be explicitly calculated. (i) Full circle θ A = −π, θ B = π. In that case the Coulomb integral is known as the Morris integral [70] leading to where M(n, a, b) is defined Eq (14) in [18] . This result is valid in the high temperature phase with nb 2 < 1. From this expression for integer moments there is a well defined procedure to obtain the double sided Laplace transform of the free energy first in the high temperature phase b < 1 via an analytic continuation. Defining t = −bn we obtain is the generalized Barnes function, see Eq. (44) in [39] and [71] . We note that if we multiply both sides of the equation by Γ(1 + t b ), the right hand side (without the factor According to the freezing duality conjecture [19, 39] we obtain the double sided Laplace transform (DSLT) in the low temperature phase b > 1. The result can be written as where the r.h.s. is our main result, i.e. the DSLT of the PDF of δN m for the full circle which, according to (16) , is the b → +∞ limit of the l.h.s of (25) . Here c = 3 2 log log(N/r) + c and c is a constant that we cannot determine by this method. Expansion of Eq. (26) around t = 0 leads to the large N asymptotics (4)-(6) for the cumulants, together with the predicted values for the coefficientsC (β) k in (7) and C k (2π) in (9). The C k (2π) equal, up to a factor (−1) k , the cumulants C k given in [18] for the fBm0 bridge, checked against numerics there for k = 2, 3, 4. These coefficients are studied in more details in [24] . (ii) maximum over a mesoscopic interval 1 N 2π Let us set φ a = θ A + x a , with x a ∈ [0, 1], and recall = θ B − θ A . In the limit of small 1 Eq (21) gives One recognizes now the Selberg integral [70] in the form which arises in the study of the fBm0 on an interval [18, 19] . Using the known expression for its analytical continuation (see Eq. (239) in [19] ) and following the same steps as above, we obtain the DSLT in the high temperature phase b < 1 with t = −nb as whereK β (b) = K β (b)(2π) β−1 . The duality invariance can be similarly checked and from the FDC we find that the DSLT in the low temperature phase b > 1 is given again by (25) with our second main result, i.e. the DSLT of the PDF of δN m for the small interval where c = 3 2 log log N +c . Expansion around t = 0 leads to the same coefficientsC (β) k which are thus independent of (as can be seen already from (17)) and to the result for C k ( ) in (10) , again related to the ones for the fBm0 on an interval given and numerically checked in [18] . The formulas (24), (29) and (28), (26) for the DSLT were obtained for real values of the parameter t. We expect them to extend to a domain around t = 0 in the complex plane. For real t this domain cannot contain t = Q/2 for β > 1 and t = 1 for β ≤ 1, which is the location of a termination point transition for the pinned fBm0 (it corresponds to events when the minimum is at θ m ≈ θ A ), analyzed in [18, 42] . The domain should also be contained within Im(t) < 1/2 (say for β = 2) because of the integer nature of the field N (θ). Extending the formula beyond remains open [72] . As to the position of the maximum θ m ∈ [θ A , θ B ], we recall that its statistics for the fBm0 on an interval has been investigated in [19] by calculating those of the Jacobi ensemble of random matrices and performing the continuation to n = 0. Defining y m = (θ m − θ A )/ , the moments E(y k m ) are thus the ones given in [19] (in Eqs. (129-130) for k = 2, 4 and Eqs. (101),(98-100), and Appendix C for general k). Extending that calculation to treat the case of the mesoscopic interval, one checks that the additional factors in (27) do not contribute, and arrives at the results mentioned. In conclusion, we obtained the cumulants of the maximum of the deviation of the counting function from its mean on an interval, for eigenvalues of random unitary matrices and for free and interacting fermions on the circle. They inherit features both from the fBm0 log-correlated field and from the fermionic full counting statistics. The results for the mesoscopic interval are expected to be universal for a broader class of random matrix ensembles, as well as for fermions on a lattice in the dilute limit (and possibly beyond [73] as remains to be investigated). It opens the way to study extremal statistics of counting functions in 1D or 2D domains, for other determinantal processes which often relate to the Gaussian free field. It should predict the extremal statistics of counting staircases associated to the zeroes of the Rieman zeta function on the critical line. Finally, as was mentioned, the distribution of δN m provides the first step towards studying such important characteristics of the counting function as the Kolmogorov-Smirnov statistics. It is necessary to mention that though the minimum and the maximum for the process δN (θ) on the same interval are identically distributed (up to sign reversal), in general the min and max values for log-correlated processes are nontrivially correlated [74], so our results are yet insufficient for characterizing the KS-like statistics. This remains an interesting challenge for the future studies. 2 ) G β (z) was used, however one can check that for Eq. (24) it is immaterial (see also Section 13.2 in [19] ). [72] Results from [2, 67] for β = 2, suggest that, treating nb = −t and b as independent variables, the integrand in (21) can be extended formally to a sum over bn → bn+iZ, b → b + iZ. We provide some additional details for some of the calculations described in the manuscript of the Letter. (β) k Let us recall the formula given in the text for the coefficientsC (β) k which enter in the cumulants of the PDF for δN m , namely, for β = 2s/r, with s, r mutually prime and k ≥ 2 To obtain more explicit expressions we use that for k ≥ 2 where ψ (k) (x) = d k+1 dx k+1 log Γ(x). Hence, for even k = 2p, defining p = s − q we obtaiñ and we recall that odd cumulants vanish. Since any real β can be reached by a sequence β = 2s n /r n of arbitrary large s n , r n we can obtain an alternative expression valid for ant β in terms of a convergent infinite series. We need to distinguish two cases: Cumulants C 2p with p ≥ 2. In that case we see that the large s, r behavior in (33) is dominated by the divergence of φ k (x) near x = 0. We use that One finds for k = 2p with p ≥ 2C One of the sum can be carried out leading to two equivalent "dual" expressions where we have used that ψ (2p−1) (1) = (2p − 1)!ζ(2p). The above series are convergent for p ≥ 2, since at large x one has ψ (2p−1) (x) Hence the result is analytic in β > 0. This asymptotics can be used to obtain the large β expansionC as well as the small β expansionC As an example we give more explicitly the fourth cumulant One can then check that this formula, valid for any β, correctly reproduces for the cases β = 2s/r, with s, r mutual primes, the same result as the original formula (33) , for instance one finds Let us also give more detailed asymptotics at large and small β The fourth cumulant is plotted as a function of β in the Figure 1 , together with the large and small β asymptotics which, as we see, are quite accurate. (β) 2 . The second cumulant reads, for β = 2s/r To study the limit where both r, s → +∞ with a fixed (more precisely, converging) ratio β = 2s/r, it is useful to decompose φ 2 (x) = − 1 x 2 +φ 2 (x), whereφ 2 (x) is regular at x = 0, and to introduce r−1 ν=0 1 1+ν = H r log r + γ E + O(1/r). Then one has in that limit Hence need to evaluate the limit where we have used that where the second line is obtained writing p = r + ν and using ψ (1) (x) ∼ 1/x at large x, but the full equivalence has also been confirmed numerically. Hence we can take the large r, s limit in (45) , the factors log 2 cancel, and we finally obtain the second cumulant for any β as the following convergent "dual" series Note the non trivial term 2 log(β/2) in the last expression, arising from the replacement −2 log r = −2 log s+2 log(β/2) in (43) . For β = 2 one recoversC (β=2) 2 = 2 + 2γ E . We also find either from (48) , or from the original formula (43) One obtains the series at large and small β Note that the leading term agrees with (43) although that result assumed p ≥ 2. The second cumulant is plotted as a function of β in the Figure 1 together with the large and small β asymptotics which, as we see, are again quite accurate. (48) and (51) . Let us recall the result given in the text for the amplitudes C k ( ) for = 2π and 1. For any k ≥ 2 We now use Eq. (32), and we also use that ψ (k) (1) = (−1) k+1 k!ζ(k + 1) and ψ (k) (2) = ψ (k) (1) + (−1) k k! and ψ (k) (4) = ψ (k) (1) + (−1) k k!(1 + 2 −k−1 + 3 −k−1 ). We obtain for the full circle and for the mesoscopic interval 1 Let us demonstrate that naively replacing the difference δN θ A (θ) with its Gaussian approximation is not sufficient for the purpose of characterizing the maximum of the process. For this end, we make the corresponding replacement in the expression (first line of 17) for the integer moments of Z b , yielding where now the expectation is over the mean-zero Gaussian process W β (θ) with the covariance given by (14) and the variance E(W (θ) 2 ) = β −1 log N . Due to Gaussian nature of the process the expectation is readily taken via the identity Substituting here the value (14) for the covariance and the associated variance and recalling that 4 sin 2 θ1−θ2 2 = |e iθ1 − e iθ2 | 2 we immediately arrive at the expression for the moments: dφ a which misses exactly the factor |A β (b)| 2n |A β (bn)| 2 when compared to the formula (21) . As those factors contribute to the cumulants for the maximum of the process, the Gaussian approximation is clearly insufficient for this purpose. Preliminary remark. Consider two random variables X 1 and X 2 . By definition the connected moments (also called bivariate cumulants) are given by Let us define the following biased average Expanding the r.h.s of (59) in powers of t 2 we see that and so on, which is also equivalent (upon multiplying by 1/q! and summing over q) to the following formula for the generating functions of the bivariate cumulants of lowest order in X 2 which will be useful below. Let us discuss first the mesoscopic interval. Let us denote, as in the text, y = θ−θ A ∈ [0, 1], and y k the k-th moment of the random variable y with respect to the Gibbs measure associated to Z b defined in (15) , for a fixed random configuration of the eigenvalues θ i . One can evaluate the following ratio of averages w.r.t. the measure CUE β for the eigenvalues The numerator in the l.h.s. of (63) equals Eq. (27) of the text with x a → y a and y k 1 inserted in the integrand. The corresponding ratio is thus the k-th moment of the Jacobi ensemble denoted y k β,a,b,n in [19] with the identification of parameters corresponding to fBm0 (see Eqs. (56, 57) there). Note that the extra factors containing A β (z) in (27), not present in the fBm0, drop out in the ratio. The expression for the y k β,a,b,n were obtained in [19] and we denote M k (t = −bn, b) these expressions, which are rational fractions of the variables t = −bn and b. We thus obtain which is valid in the high temperature phase b < 1. The simplest examples are the first two moments k = 1, 2. From (107) and (190) in [19] we obtain These expressions are duality invariant, i.e. does not change under b → 1/b. All moments M k (t, b) share this property [19] (their explicit expressions are given in (91-92) there). Hence the freezing duality conjecture (FDC) allows to continue (64) for b > 1. As in the text, the r.h.s. is duality invariant if multiplied by Γ(1 + t b ), hence the value of the l.h.s, as a function of b, freezes at b = 1. Taking b → +∞ we obtain where y m is the position of the maximum. Setting t = 0 yields the results for the moments E(y k m ) quoted in the text. Let us denote the centered variables Consider (66) for k = 1. Using that E(y m ) = 1 2 , this can be written as where it is useful to define the following averages which represent averages under a biased probability e −2π In particular The result for p = 1 shows that positions of maximum y m > 1/2 correlate with values of the maximum larger than the average, consistent with the pinning at y = 0, i.e. δN (θ = θ A ) = 0, while the boundary condition at y = 1 is free. Since δN m − E(δN m ) is typically ∼ √ log N the correlation with the Gaussian part of the fluctuations of the value of the maximum is absent in the correlation for p = 1 (which is O(1)). Now, it is easy to see that (70) and (68) imply that all higher bi-variate cumulants vanish, i.e. the information contained in (68) can be summarized as For t = 0 we obtain the result given in the text E(y 2 m ) = 17 50 . Expansion of the first equation in powers of t allows to obtain all joint moments of the form E(y 2 m (δÑ m ) p ) using our results for the cumulants of the value of the maximum (given in the text). Alternatively we may write the bi-variate cumulants (see preliminary remark above) Expanding in powers of t on both sides we see that for k = 2, only the first three connected moments are non zero. Consider now the average of cos kφ with respect to the Gibbs measure associated to Z b , defined in (15), on the full circle, i.e. with θ A = −π and θ B = π. Again one can evaluate the ratio of averages w.r.t. the measure CUE β for the = cos(kφ) β,µ,n = (−1) k y k β,a,b,n | (β,a,b)→(b,−1−b 2 ,2nb 2 ) =M k (t = −bn, b) The numerator in the l.h.s. of (76) equals Eq. (21) of the text with cos kφ 1 inserted in the integrand. The second equality is the conjecture obtained in Eqs. (157-158) in [19] (where cos(kφ) β,µ,n denotes the l.h.s. in (158) there, with κ = −β 2 → −b 2 and µ → nb 2 ) which relates the moments in the circular ensemble to those on the interval, i.e the moments y k β,a,b,n already used in the previous section. The different specialisations of the parameters leads to other rational functionsM k (t = −bn, b). From (107) and (91-92) in [19] we obtaiñ which, upon expanding in t yields all joint moments of the form E(cos(2θ m )(δÑ m ) p ). Charge distribution in quantum shot noise Quantum fluctuations of one-dimensional free fermions and Fisher-Hartwig formula for Toeplitz determinants Characterizing correlations with full counting statistics: Classical Ising and quantum XY spin chains Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas Luttinger liquids with multiple Fermi edges: Generalized Fisher-Hartwig conjecture and numerical analysis of Toeplitz determinants Universality in the full counting statistics of trapped fermions Phase transitions and edge scaling of number variance in Gaussian random matrices Non-interacting fermions at finite temperature in a ddimensional trap: universal correlations Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory Random matrices and entanglement entropy of trapped Fermi gases Full Counting Statistics in a Propagating Quantum Front and Random Matrix Spectra Statistical Theory of the Energy Levels of Complex Systems I Matrix Models for Circular Ensembles Exact Results for a Quantum Many-Body problem in One Dimension On Cramer-von Mises statistic for the spectral distribution of random matrices Fractional Brownian motion with Hurst index H = 0 and the Gaussian Unitary Ensemble Logcorrelated random-energy models with extensive freeenergy fluctuations: Pathologies caused by rare events as signatures of phase transitions Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes Polymers on disordered trees, spin glasses, and travelling waves Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal 1/f noise Convergence of the centered maximum of log-correlated Gaussian fields M where Ui is a standard centered log-correlated field. Clearly the position of the maximum for Ui, hence of the one for Vi On the distribution of spacings between zeros of the zeta function Random matrix theory and ζ(1/2 + it) Random matrices and number theory: some recent themes Mesoscopic fluctuations of the zeta zeros Strong Szego asymptotics and zeros of the zeta-function On the characteristic polynomial of a random unitary matrix On the maximum of the CβE field Sur le chaos multiplicatif Localization in two dimensions, gaussian field theories, and multifractality Multifractality and freezing phenomena in random energy landscapes: An introduction Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity Integrability of Liouville theory: proof of the DOZZ Formula Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields On Barnes beta distributions and applications to the maximum distribution of the 2D Gaussian Free Field A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications Operator Product Expansion in Liouville Field Theory and Seiberg type transitions in log-correlated Random Energy Models Liouville field theory and log-correlated Random Energy Models One step replica symmetry breaking and extreme order statistics of logarithmic REMs Glassy phase and freezing of log-correlated Gaussian potentials An elementary approach to Gaussian multiplicative chaos On the distribution of the maximum value of the characteristic polynomial of GUE random matrices Maximum of the characteristic polynomial of random unitary matrices The Maximum of the CUE Field Freezing and decorated Poisson point processes The Fyodorov-Bouchaud formula and Liouville conformal field theory The distribution of Gaussian multiplicative chaos on the unit interval Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos -the L 2 -phase Random Hermitian matrices and Gaussian multiplicative chaos Freezing transitions, characteristic polynomials of random matrices, and the Riemann zeta-function Freezing transitions and extreme values: random-matrix theory, ζ( 1 2 + it) and disordered landscapes Maxima of a randomized Riemann zeta function, and branching random walks Maximum of the Riemann zeta function on a short interval of the critical line On the extreme values of the Riemann zeta function on random intervals of the critical line Moments of the Riemann zeta function on short intervals of the critical line On the partition function of the Riemann zeta function, and the Fyodorov-Hiary-Keating conjecture On the circle, GM C γ = lim CβEn for γ = 2/β Applications and generalizations of Fisher-Hartwig asymptotics Toeplitz determinants: some applications, theorems, and conjectures = det 1≤j,k≤N Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities Ann 22) each factor |G(1 + iγ)| 2 is meant as G(1 + iγ)G(1 − iγ), valid more generally for γ in the complex plane Full counting statistics in the Haldane-Shastry chain The importance of the Selberg integral Acknowledgments: We thank X. Cao for a very inspiring collaboration on the topic of pinned fBm0 and J.P. Keating for an insightful discussion of the early version of this paper. YVF thanks the Philippe Meyer Institute for Theoretical Physics at ENS in Paris. PLD acknowledges support from ANR grant ANR-17-CE30-0027-01 RaMa-TraF.