Statistics of extremes in eigenvalue-counting staircases
SStatistics of extremes in eigenvalue-counting staircases
Yan V. Fyodorov
1, 2 and Pierre Le Doussal King’s College London, Department of Mathematics, London WC2R 2LS, United Kingdom L.D.Landau Institute for Theoretical Physics, Semenova 1a, 142432 Chernogolovka, Russia Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, PSL Research University,CNRS, Sorbonne Universités, 24 rue Lhomond, 75231 Paris, France
We consider the number N θ A ( θ ) of eigenvalues e iθ j of a random unitary matrix, drawn fromCUE β ( N ) , in the interval θ j ∈ [ θ A , θ ] . The deviations from its mean, N θ A ( θ ) − E ( N θ A ( θ )) , form arandom process as function of θ . We study the maximum of this process, by exploiting the mappingonto 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, weobtain the cumulants of the distribution of that maximum for any β > . It exhibits combinedfeatures of standard counting statistics of fermions (free for β = 2 and with Sutherland-type inter-action for β (cid:54) = 2 ) in an interval and extremal statistics of the fractional Brownian motion with Hurstindex H = 0 . The β = 2 results are expected to apply to the statistics of zeroes of the RiemannZeta function. PACS numbers:
Characterizing the full counting statistics of the fluc-tuations of the number N of d fermions in an inter-val is important in numerous physical contexts, both forground state and dynamical properties. It appears e.g.in shot noise [1], in fermion chains [2, 3], in interactingBose gases [4], in non-equilibrium Luttinger liquids [5],in trapped fermions [6–8], and for studying related ob-servables, such as the entanglement entropy [9–11] or thestatistics of local magnetization in quantum spin chains[12]. An equivalent problem can be formulated as count-ing eigenvalues of large random matrices (RM). As is wellknown since Dyson’s work [13], such eigenvalues behaveas classical particles with 1-d Coulomb repulsion at in-verse temperature β > . Namely, consider a unitary N × N matrix U and denote the corresponding unimod-ular eigenvalues as z j = e iθ j , j = 1 , . . . , N , with phases θ i ∈ ] − π, π ] . Then for any given β > one can constructthe so-called Circular β -Ensemble CUE β ( N ) in such away that the expectation of a function depending onlyon the eigenvalues of U will be given by E ( F ) = c N N (cid:89) j =1 (cid:90) π − π dθ i (cid:89) ≤ j
32 log log N + c ( β ) (cid:96) (4)where c ( β ) (cid:96) = O (1) is an unknown (cid:96) -dependent constant.The variance for the maximum δ N m exhibits to the lead- ing order the extensive universal logarithmic growth typ-ical for pinned log-correlated fields [19], on top of whichwe can evaluate the corrections of the order of unity: E c ( δ N m ) (cid:39) β (2 π ) (2 log N + ˜ C ( β )2 + C ( (cid:96) )) (5)Finally, the higher cumulants converge to a finite limitas N → ∞ : E c ( δ N km ) (cid:39) k/ β k/ (2 π ) k ( ˜ C ( β ) k + C k ( (cid:96) )) , (6)where the constants C k ( (cid:96) ) = O (1) depend on the length (cid:96) of the interval and will be given below in two limitingcases. The (cid:96) − independent constants ˜ C ( β ) k for k ≥ aregiven by ˜ C ( β ) k = d k dt k | t =0 log( A β ( t ) A β ( − t )) (7)where A β ( t ) = r − t / r − (cid:89) ν =0 s − (cid:89) p =0 G (1 − ps + ν + it (cid:113) β r ) G (cid:0) − ps + νr (cid:1) (8)Here G ( z ) denotes the standard Barnes function satis-fying G ( z + 1) = Γ( z ) G ( z ) , with G (1) = 1 . Note that allthe odd coefficients ˜ C ( β )2 k +1 vanish. Specifying for β = 2 ,one has A ( t ) = G (1+ it ) , leading to ˜ C (2)2 = 2(1+ γ E ) and ˜ C (2)4 = − ζ (3) . Notably, using (7), (8), we were able toobtain a formula for the ˜ C ( β ) k as single infinite series [27],which shows that they are smooth as a function of theDyson parameter β , thus relaxing the assumption of ra-tionality. As discussed below, the factors A β ( t ) , hence ˜ C ( β ) k , are intimately but non-trivially related to the cu-mulants of the number of fermions (free for β = 2 andwith Sutherland-type interaction for β (cid:54) = 2 ) in a meso-scopic interval of the circle.By contrast the factors C k ( (cid:96) ) are β -independent andoriginate from the problem of the maximum of a fBm0on the interval [ θ A , θ B ] . For the (cid:96) -dependent constantswe obtain explicit formula in two cases: (i) maximum over the full circle (cid:96) = 2 π . In that case [ θ A , θ B ] =] − π, π ] and we find for any k ≥ C k (2 π ) = ( − k d k dt k | t =0 log (cid:20) Γ(1 + t ) G (2 − t ) G (2 − t ) G (2 + t ) (cid:21) (9)which is related to the fBm0 bridge on ] − π, π ] studiedin [19] (ii) maximum over a mesoscopic interval N (cid:28) (cid:96) (cid:28) .For k ≥ we obtain in this regime C k ( (cid:96) ) (cid:39) (cid:96) δ k, (10) +( − k d k dt k | t =0 (cid:20) t ) G (2 − t ) G (2 + t ) G (2 − t ) G (4 − t ) (cid:21) (11)This result is related to the fBm0 on an interval, with onepinned and one free end, studied in [19]. Note that thevariance depends logarithmically on (cid:96) at small (cid:96) , whereashigher cumulants have limits as (cid:96) → . Note that l → limit is expected to provide the L (cid:29) asymptoticfor statistics of the maximum of N θ A ( θ ) in intervals ofthe order πL/N , comparable with the mean eigenvaluespacing. The universal statistics of CUE β eigenvaluesat such local scales is described by the so called sine - β process [25] and the associated counting function hasbeen studied in [26].Finally, addressing the question of the location of themaximum in (3), θ m ∈ [ θ A , θ B ] , let us define y m =( θ m − θ A ) /(cid:96) . For the mesoscopic interval, we predict thePDF of y m to be symmetric around , with E ( y m ) = and E ( y m ) = , thus deviating from the uniform dis-tribution. For the full circle we find a uniform distribu-tion for θ m [28]. However, joint moments for the position and value of the maximum show the effect of pinning at θ = θ A (see details in [27]).To elucidate the relation to fBm0, let us recall that theprocess δN θ A ( θ ) is exactly given by the difference [27] δ N θ A ( θ ) = 1 π Im log ξ N ( θ ) − π Im log ξ N ( θ A ) (12)where ξ N ( θ ) = det(1 − e − iθ U ) is the characteristic poly-nomial (CP). As shown in [29] for β = 2 (see [30] for gen-eral β > ) the joint probability density of Im log ξ N ( θ ) at two distinct points θ (cid:54) = θ converges as N → + ∞ to that of a Gaussian process W β ( θ ) of zero mean andcovariance E ( W β ( θ ) W β ( θ )) = − β log (cid:20) (cid:18) θ − θ (cid:19)(cid:21) (13)a particular instance of the 1D log-correlated Gaussianfield. Since (12) implies that δ N θ A ( θ = θ A ) = 0 in any re-alization, the relevant object is the pinned log-correlatedprocess closely related to fBm0. The log-correlated fieldsbeing highly singular always require a regularization tostudy their value distribution. The imaginary partsof the log ξ N ( θ ) for N (cid:29) provides such a naturalregularization [30, 52–54], being asymptotically a ran-dom process W which shares the covariance (13) butwith a finite variance E ( W ( θ ) ) = β − log N + O (1) .Via (12) this provides the well-known asymptotic ofthe eigenvalues/fermions number variance: E ( δ N ( θ )) (cid:39) βπ log N . We shall see however [27] that naively replac-ing the difference δ N θ A ( θ ) with its Gaussian approxima-tion π [ W β ( θ ) − W β ( θ A )] (related to the bosonization ofthe fermionic problem) is not sufficient for characterizingthe maximum of the process.Gaussian fields characterized by a logarithmic covari-ance appear in chaos and turbulence [31], branching ran-dom walks and polymers on trees [21, 22], multifrac-tal disordered systems [32, 33], two-dimensional grav-ity [34, 35]. Early works on their extrema revealed a con-nection to a remarkable freezing transition [21, 22, 32].Through exact solutions, it led to predictions for the PDFof the maximum value of a log-correlated field on the cir- cle and on the interval [36, 37], involving the freezingduality conjecture (FDC) (see [20] for an extensive discus-sion). This led to further results in theoretical and math-ematical physics [23, 38–42] and probability [24, 43–51].A log-correlated context of random CP attracting a lotof attention [30, 45–47, 55–62], none of these studies yetaddressed the eigenvalue/zeros counting function in theintervals (cid:96) = O (1) .To study the maximum of the random field δ N ( θ ) wefollow [20, 36, 37, 55, 56] and introduce a statistical me-chanics problem of partition sum: Z b = N π (cid:90) θ B θ A dφ e πb √ β/ δ N θA ( φ ) , (14)The “ inverse temperature" is equal to − πb (cid:112) β/ , andwe choose b > since we are studying here the maximumretrieved from the free energy F for b → + ∞ as δ N m = lim b → + ∞ F , F = 12 πb (cid:112) β/ Z b (15)To study the statistics of the associated free energy westart with considering the integer moments of Z b givenby E [ Z nb ] = (cid:18) N π (cid:19) n (cid:90) θ B θ A e − b √ β/ (cid:80) na =1 N ( φ a − θ A ) n (cid:89) a =1 dφ a × E [ N (cid:89) j =1 e πb √ β/ (cid:80) na =1 ( χ ( φ a − θ j ) − χ ( θ A − θ j )) ] (16)The expectation value in (16) over the CUE β ( N ) com-puted using (1) has the form E [ (cid:81) Nj =1 g ( θ j )] where we de-fined log g ( θ ) = 2 πb (cid:112) β/ n (cid:88) a =1 ( χ ( φ a − θ ) − χ ( θ A − θ )) (17)This can be further rewritten for any φ a , θ, θ A ∈ ] − π, π ] with φ a > θ A as log g ( θ ) = b (cid:112) β/ n (cid:88) a =1 φ a − nθ A (18) + n arg e i ( θ A − θ + π ) − n (cid:88) a =1 arg e i ( φ a − θ + π ) ] where we define the arg function as arg e iφ = (cid:40) φ − π < φ ≤ πφ − π π < φ ≤ π (19)For β = 2 , E [ (cid:81) Nj =1 g ( θ j )] = det ≤ j,k ≤ N [ g j − k ] is aToeplitz determinant, where g p = (cid:82) π − π dθ π e − ipθ g ( θ ) is theassociated symbol, and g ( θ ) according to (18)-(19) hasjump singularities. The corresponding asymptotics as N → ∞ is given by the famous Fisher-Hartwig (FH)formula [64] proved rigorously in [65]. For a generalrational β extension of FH formula has been conjecturedin [63]. Specifying the expressions in [63] to our casegives for N → + ∞ and nb < E [ Z nb ] (cid:39) (cid:18) N π (cid:19) n N b ( n + n ) | A β ( b ) | n | A β ( bn ) | × (cid:90) θ B θ A (cid:89) ≤ a
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Statistics of extremes in eigenvalue-counting staircases
Yan V. Fyodorov and P. Le Doussal
We provide some additional details for some of the calculations described in the manuscript of the Letter.
CUMULANT AMPLITUDES ˜ C ( β ) k AS A FUNCTION OF β Let us recall the formula given in the text for the coefficients ˜ C ( β ) k which enter in the cumulants of the PDF for δ N m , namely, for β = 2 s/r , with s, r mutually prime and k ≥ C ( β ) k = d k dt k | t =0 log( A β ( t ) A β ( − t )) (27) A β ( t ) = r − t / r − (cid:89) ν =0 s − (cid:89) p =0 G (1 − ps + ν + it (cid:113) β r ) G (cid:0) − ps + νr (cid:1) (28)To obtain more explicit expressions we use that for k ≥ d k dy k log G ( x + y ) | y =0 = φ k ( x ) := ( k − ψ ( k − ( x ) + ( x − ψ ( k − ( x ) − δ k, (29)where ψ ( k ) ( x ) = d k +1 dx k +1 log Γ( x ) . Hence, for even k = 2 p , defining p = s − q we obtain ˜ C ( β )2 p = ( − p rs ) p r − (cid:88) ν =0 s (cid:88) q =1 φ p ( νr + qs ) − rδ p, (30)and we recall that odd cumulants vanish.Since any real β can be reached by a sequence β = 2 s n /r n of arbitrary large s n , r n we can obtain an alternativeexpression valid for any β in terms of a convergent infinite series. We need to distinguish two cases: Cumulants C p with p ≥ . In that case we see that the large s, r behavior in (30) is dominated by the divergenceof φ k ( x ) near x = 0 . We use that φ p ( x ) = − (2 p − x p + O (1) (31)One finds for k = 2 p with p ≥ C ( β )2 p = ( − p +1 p − ∞ (cid:88) ν =0 ∞ (cid:88) q =1 ν (cid:113) β + q (cid:113) β ) p (32)One of the sum can be carried out leading to two equivalent "dual" expressions ˜ C ( β )2 p = ( − − p β p ∞ (cid:88) ν =0 ψ (2 p − (1 + βν − p +1 β p ∞ (cid:88) q =1 ψ (2 p − ( 2 qβ ) (33)where we have used that ψ (2 p − (1) = (2 p − ζ (2 p ) . The above series are convergent for p ≥ , since at large x onehas ψ (2 p − ( x ) (cid:39) (2 p − z p − . Hence the result is analytic in β > . This asymptotics can be used to obtain the large β expansion ˜ C ( β )2 p = ( − − p (2 p − ζ (2 p ) β p + ( − p +1 p (2 p − ζ (2 p −
1) 1 β p − + O ( β − p ) (34)as well as the small β expansion ˜ C ( β )2 p (cid:39) ( − p +1 − p (2 p − ζ (2 p − β p − , β (cid:28) (35)As an example we give more explicitly the fourth cumulant ˜ C ( β )4 = − ∞ (cid:88) ν =0 ∞ (cid:88) q =1 ν (cid:113) β + q (cid:113) β ) = − β ∞ (cid:88) q =1 ψ (3) ( 2 qβ ) = − β ∞ (cid:88) ν =0 ψ (3) (1 + βν (36)One can then check that this formula, valid for any β , correctly reproduces for the cases β = 2 s/r , with s, r mutualprimes, the same result as the original formula (30), for instance one finds ˜ C ( β =2)4 = − ζ (3) , ˜ C ( β =1)4 = π − ζ (3) , ˜ C ( β =4)4 = − ζ (3) − π (37)Let us also give more detailed asymptotics at large and small β ˜ C ( β )4 = − π β − ζ (3) β + 4 π β − ζ (5) β + O ( β − ) (38) = − βζ (3) − π β − β ζ (5)2 + O (cid:0) β (cid:1) (39)The fourth cumulant is plotted as a function of β in the Figure 3, together with the large and small β asymptoticswhich, as we see, are quite accurate. Second cumulant ˜ C ( β )2 . The second cumulant reads, for β = 2 s/r ˜ C ( β )2 = − rs r − (cid:88) ν =0 s (cid:88) q =1 φ ( νr + qs ) − r (40)To study the limit where both r, s → + ∞ with a fixed (more precisely, converging) ratio β = 2 s/r , it is useful todecompose φ ( x ) = − x + ˜ φ ( x ) , where ˜ φ ( x ) is regular at x = 0 , and to introduce (cid:80) r − ν =0 11+ ν = H r (cid:39) log r + γ E + O (1 /r ) . Then one has in that limit − rs r − (cid:88) ν =0 s (cid:88) q =1 ˜ φ ( νr + qs ) → − (cid:90) dx (cid:90) dy ˜ φ ( x + y ) = − (cid:90) dss ˜ φ ( s ) + (cid:90) ds (2 − s ) ˜ φ ( s )) = 2 log 2 (41)Hence need to evaluate the limit ˜ C ( β )2 (cid:39) γ E + 2 r − (cid:88) ν =0 [ s (cid:88) q =1 β/ ν β + q ) −
11 + ν ] (42) = 2 log 2 + 2 γ E + 2 r − (cid:88) ν =0 [ β ψ (1) (1 + βν − β ψ (1) (1 + s + βν −
11 + ν ] (43)where we have used that (cid:80) sq =1 1( q + a ) = ψ (1) (1 + a ) − ψ (1) (1 + s + a ) . Now one can check that lim r → + ∞ r − (cid:88) ν =0 β ψ (1) (1 + β r + βν (cid:39) lim r → + ∞ r − (cid:88) p = r β + p = log 2 (44)where the second line is obtained writing p = r + ν and using ψ (1) ( x ) ∼ /x at large x , but the full equivalence hasalso been confirmed numerically. Hence we can take the large r, s limit in (42), the factors log 2 cancel, and we finallyobtain the second cumulant for any β as the following convergent "dual" series ˜ C ( β )2 = 2 γ E + 2 + ∞ (cid:88) ν =0 [ + ∞ (cid:88) q =1 β/ ν β + q ) −
11 + ν ] = 2 γ E + 2 + ∞ (cid:88) ν =0 [ β ψ (1) (1 + βν −
11 + ν ] (45) = 2 γ E + 2 log( β/
2) + 2 ∞ (cid:88) q =1 ( 2 β ψ (1) ( 2 qβ ) − q ) (46)Note the non trivial term β/ in the last expression, arising from the replacement − r = − s +2 log( β/ in (40). For β = 2 one recovers ˜ C ( β =2)2 = 2 + 2 γ E . We also find either from (45), or from the original formula (40) ˜ C ( β =1)2 = 2 + 2 γ E − π , ˜ C ( β =4)2 = 2 + 2 γ E + π (47)0One obtains the series at large and small β ˜ C ( β )2 = π β γ E − π β + 4 ζ (3)3 β − ζ (5)15 β + O ( β − ) (48) = 2 log( β/
2) + 2 γ E + π β
12 + β ζ (3)12 − β ζ (5)240 + O (cid:0) β (cid:1) (49)Note that the leading term agrees with (40) although that result assumed p ≥ .The second cumulant is plotted as a function of β in the Figure 3 together with the large and small β asymptoticswhich, as we see, are again quite accurate. β - - - - C β β - - C β Figure 3: Left: fourth cumulant amplitude ˜ C ( β )4 plotted (in blue) as a function of β from (36). Dotted and dashed lines aresmall and large β asymptotics, (38) respectively. Right: same for second cumulant amplitude ˜ C ( β )2 (in blue) from (45) and (48). CUMULANT AMPLITUDES C k ( (cid:96) ) Let us recall the result given in the text for the amplitudes C k ( (cid:96) ) for (cid:96) = 2 π and (cid:96) (cid:28) . For any k ≥ C k (2 π ) = ( − k d k dt k | t =0 log (cid:20) Γ(1 + t ) G (2 − t ) G (2 − t ) G (2 + t ) (cid:21) (50) C k ( (cid:96) ) (cid:39) (cid:96) δ k, + ( − k d k dt k | t =0 (cid:20) t ) G (2 − t ) G (2 + t ) G (2 − t ) G (4 − t ) (cid:21) (51)We now use Eq. (29), and we also use that ψ ( k ) (1) = ( − k +1 k ! ζ ( k + 1) and ψ ( k ) (2) = ψ ( k ) (1) + ( − k k ! and ψ ( k ) (4) = ψ ( k ) (1) + ( − k k !(1 + 2 − k − + 3 − k − ) . We obtain for the full circle C (2 π ) = π , C (2 π ) = 2 π − ζ (3) , C (2 π ) = 1415 π − ζ (3) (52) C k (2 π ) = (cid:0)(cid:0) − ( − k + 3( − k (cid:1) ζ ( k −
1) + (cid:0) − k − − k (cid:1) ζ ( k ) (cid:1) Γ( k ) , k ≥ and for the mesoscopic interval (cid:96) (cid:28) C ( (cid:96) ) = 2 log (cid:96) + 94 , C ( (cid:96) ) = −
174 + 8 π − ζ (3) , C ( (cid:96) ) = 998 + 45 π − ζ (3) (53) C k ( (cid:96) ) = 6 − k (cid:0)(cid:0) − ( − k + ( − k k +1 + 2 k +1 k (cid:1) ζ ( k −
1) + ( − k (cid:0) k − (cid:1) ζ ( k ) + ( − k (cid:0) k +1 + 1 (cid:1)(cid:1) Γ( k ) , k ≥ DISTRIBUTION OF THE MAXIMUM OVER A MESOSCOPIC INTERVAL N (cid:28) (cid:96) (cid:28) π Here we sketch the derivation of our second main result, (26).To this end we set φ a = θ A + (cid:96)x a , with x a ∈ [0 , , and recall (cid:96) = θ B − θ A . Eq (20) gives E [ Z nb ] (cid:39) (cid:18) N (cid:96) π (cid:19) n ( N (cid:96) ) b ( n + n ) | A β ( b ) | n | A β ( bn ) | n (cid:89) a =1 (cid:90) dx a (cid:89) ≤ a
Our starting point is the characteristic polynomial defined in the text, which we rewrite ξ N ( θ ) = (cid:89) j (cid:16) − e i ( θ j − θ ) (cid:17) = (cid:89) j e i ( θ j − θ − π ) / θ j − θ e − i N ( π + θ )2 + i (cid:80) Nj =1 θ j N (cid:89) j =1 sgn (cid:20) sin θ j − θ (cid:21) N (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) sin θ j − θ (cid:12)(cid:12)(cid:12)(cid:12) (56)Further using N (cid:89) j =1 sgn (cid:20) sin θ j − θ (cid:21) = N (cid:89) j =1 sgn ( θ j − θ ) = ( − θ j <θ ) = e iπ θ j <θ ) where θ j < θ ) := (cid:80) Nj =1 χ ( θ − θ j ) is the number of θ j not exceeding the value θ , we see that Im log ξ N ( θ ) = − N π + θ ) + 12 (cid:88) j =1 θ j + π θ j < θ ) (57)implying via the definition (2) π Im log ξ N ( θ ) − π Im log ξ N ( θ A ) = − N π ( θ − θ A ) + θ j < θ ) − θ j < θ A ) := δ N θ A ( θ ) (58)exactly as claimed in (12). DEFICIENCY OF THE LOG-CORRELATED GAUSSIAN APPROXIMATION FOR CHARACTERISINGTHE MAXIMUM OF δ N θ A ( θ ) . Let us demonstrate that naively replacing the difference δ N θ A ( θ ) with its Gaussian approximation π [ W β ( θ ) − W β ( θ A )] 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 16) for the integer moments of Z b , yielding E [ Z nb ] = (cid:18) N π (cid:19) n (cid:90) θ B θ A E (cid:104) e b √ β/ ( (cid:80) na =1 W β ( φ a ) − nW β ( θ A ) ) (cid:105) n (cid:89) a =1 dφ a (59)where now the expectation is over the mean-zero Gaussian process W β ( θ ) with the covariance given by (13) andthe variance E ( W ( θ ) ) = β − log N . Due to Gaussian nature of the process the expectation is readily taken via theidentity E (cid:104) e b √ β/ ( (cid:80) na =1 W β ( φ a ) − nW β ( θ A ) ) (cid:105) = e βb [ n ( n +1) E ( W ( θ A ) ) +2 (cid:80) na The formulas (23), (25), (26), of the main text 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/ for b > and t = 1 for b ≤ , which is the location of a termination point transition for the pinned fBm0 (itcorresponds to events when the minimum is at θ m ≈ θ A ), analyzed in [19, 40]. The domain should also be containedwithin Im( t ) < / (for β = 2 ) because of the integer nature of the field N ( θ ) . Extending the formula beyond remainsopen. However, results from [2, 65] for β = 2 , suggest that, treating nb = − t and b as independent variables, theintegrand in (20) can be extended formally to a sum over bn → bn + i Z , b → b + i Z . Investigating these properties isleft for future studies. MOMENTS OF THE POSITION OF THE MAXIMUM 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 beeninvestigated in [20] by calculating those of the Jacobi ensemble of random matrices and performing the continuationto n = 0 . Defining y m = ( θ m − θ A ) /(cid:96) , the moments E ( y km ) are thus the ones given in [20] (in Eqs. (129-130) for k = 2 , and Eqs. (101),(98-100), and Appendix C for general k ). Extending that calculation to treat the case ofthe mesoscopic interval, one checks that the additional factors in (54) do not contribute, and arrives at the resultsmentioned in the Letter. JOINT MOMENTS OF THE POSITION AND THE VALUE OF THE MAXIMUM Preliminary remark . Consider two random variables X and X . By definition the connected moments (also calledbivariate cumulants) are given by E c ( X q X q ) = ∂ q t | t =0 ∂ q t | t =0 log E ( e t X + t X ) (61)Let us define the following biased average (cid:104) f ( X ) (cid:105) t = E ( f ( X ) e t X ) E ( e t X ) (62)Expanding the r.h.s of (61) in powers of t we see that E c ( X X q ) = ∂ qt | t =0 (cid:104) X (cid:105) t , E c ( X X q ) = ∂ qt | t =0 ( (cid:104) X (cid:105) t − (cid:104) X (cid:105) t ) (63)and so on, which is also equivalent (upon multiplying by /q ! and summing over q ) to the following formula for thegenerating functions of the bivariate cumulants of lowest order in X E c ( X e t X ) = (cid:104) X (cid:105) t = E ( X e t X ) E ( e t X ) , E c ( X e t X ) = (cid:104) X (cid:105) t − (cid:104) X (cid:105) t (64)which will be useful below.3 Mesoscopic interval Let us discuss first the mesoscopic interval. Let us denote, as in the text, y = θ − θ A (cid:96) ∈ [0 , , and (cid:104) y k (cid:105) the k -thmoment of the random variable y with respect to the Gibbs measure associated to Z b defined in (14), for a fixedrandom configuration of the eigenvalues θ i . One can evaluate the following ratio of averages w.r.t. the measure CUE β for the eigenvalues E [ (cid:104) y k (cid:105) Z nb ] E [ Z nb ] = (cid:104) y k (cid:105) β,a,b,n | ( β,a,b ) → ( b, nb , = M k ( t = − bn, b ) (65)The numerator in the l.h.s. of (65) equals Eq. (54) of the text with x a → y a and y k inserted in the integrand. Thecorresponding ratio is thus the k -th moment of the Jacobi ensemble denoted (cid:104) y k (cid:105) β,a,b,n in [20] with the identificationof parameters corresponding to fBm0 (see Eqs. (56,57) there). Note that the extra factors containing A β ( z ) in (54),not present in the fBm0, drop out in the ratio. The expression for the (cid:104) y k (cid:105) β,a,b,n were obtained in [20] and we denote M k ( t = − bn, b ) these expressions, which are rational fractions of the variables t = − bn and b . We thus obtain E ( (cid:104) y k (cid:105) e − π √ β t F ) = M k ( t, b ) E ( e − π √ β t F ) (66)which is valid in the high temperature phase b < . The simplest examples are the first two moments k = 1 , . From(107) and (190) in [20] we obtain M ( t, b ) = 12 − tb b ) , M ( t, b ) = (cid:0) b − bt + 1 (cid:1) (cid:0) b (cid:0) b (cid:0) b − bt + t + 9 (cid:1) − t (cid:1) + 4 (cid:1) b + 19 b + 19 b + 6) (67)These expressions are duality invariant, i.e. does not change under b → /b . All moments M k ( t, b ) share this property[20] (their explicit expressions are given in (91-92) there). Hence the freezing duality conjecture (FDC) allows tocontinue (66) for b > . As in the text, the r.h.s. is duality invariant if multiplied by Γ(1 + tb ) , hence the value of thel.h.s, as a function of b , freezes at b = 1 . Taking b → + ∞ we obtain E ( y km e − π √ β tδ N m ) = M k ( t, E ( e − π √ β tδ N m ) (68)where y m is the position of the maximum. Setting t = 0 yields the results for the moments E ( y km ) quoted in the text.Let us denote the centered variables ˜ y m = y m − E ( y m ) , δ ˜ N m = δ N m − E ( δ N m ) (69)Consider (68) for k = 1 . Using that E ( y m ) = , this can be written as E (˜ y m e − π √ β tδ ˜ N m ) = − t E ( e − π √ β tδ ˜ N m ) ⇔ (cid:104) ˜ y m (cid:105) t = − t (70)where it is useful to define the following averages (cid:104) f (˜ y m ) (cid:105) t := E ( f (˜ y m ) e − π √ β tδ ˜ N m ) E ( e − π √ β tδ ˜ N m ) (71)which represent averages under a biased probability e − π √ β tδ ˜ N m × P ( δ ˜ N m ) , where P is the PDF of δ ˜ N m . Here t < corresponds to biasing the values of the maximum towards the positive values and leads to positive values on averagefor ˜ y m . Expansion in powers of t in (70) yields the relations, valid for any p ≥ E (cid:0) ˜ y m ( δ ˜ N m ) p (cid:1) = p π (cid:112) β/ E (cid:0) ( δ ˜ N m ) p − (cid:1) = p E (cid:0) ˜ y m δ ˜ N m (cid:1) × E (cid:0) ( δ ˜ N m ) p − (cid:1) (72)In particular E (cid:0) ˜ y m ( δ ˜ N m ) p (cid:1) = 1(2 π (cid:112) β/ p × , p = 10 , p = 2 (2 log N (cid:96) + ) , p = 3( − + π − ζ (3)) , p = 4 (73)The result for p = 1 shows that positions of maximum y m > / correlate with values of the maximum larger thanthe average, consistent with the pinning at y = 0 , i.e. δ N ( θ = θ A ) = 0 , while the boundary condition at y = 1 is free.Since δ N m − E ( δ N m ) is typically ∼ √ log N (cid:96) the correlation with the Gaussian part of the fluctuations of the value ofthe maximum is absent in the correlation for p = 1 (which is O (1) ). Now, it is easy to see that (72) and (70) imply4that all higher bi-variate cumulants vanish , i.e. the information contained in (70) can be summarized as E (cid:0) ˜ y m δ ˜ N m (cid:1) = 14 12 π (cid:112) β/ (74) E c (cid:0) ˜ y m ( δ ˜ N m ) p (cid:1) = 0 , p ≥ (75)consistent with (73) which is the sum of all disconnected averages.For k = 2 we obtain E ( y m e − π √ β tδ ˜ N m ) = 1100 (2 − t ) (cid:0) − t + t (cid:1) E ( e − π √ β tδ ˜ N m ) ⇔ (cid:104) y m (cid:105) t = 1100 (2 − t ) (cid:0) − t + t (cid:1) (76)For t = 0 we obtain the result given in the text E ( y m ) = . Expansion of the first equation in powers of t allows toobtain all joint moments of the form E ( y m ( δ ˜ N 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) E c ( y m e − π √ β tδ ˜ N m ) = (cid:104) y m (cid:105) t − (cid:104) y m (cid:105) t = 1400 (4 − t )(9 + 4 t ) (77)Expanding in powers of t on both sides we see that for k = 2 , only the first three connected moments are non zero. Full circle Consider now the average of cos kφ with respect to the Gibbs measure associated to Z b , defined in (14), on the fullcircle, 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 theeigenvalues E [ (cid:104) cos kφ (cid:105) Z nb ] E [ Z nb ] = (cid:104) cos( kφ ) (cid:105) β,µ,n = ( − k (cid:104) y k (cid:105) β,a,b,n | ( β,a,b ) → ( b, − − b , nb ) = ˜ M k ( t = − bn, b ) (78)The numerator in the l.h.s. of (78) equals Eq. (20) of the text with cos kφ inserted in the integrand. The secondequality is the conjecture obtained in Eqs. (157-158) in [20] (where (cid:104) cos( kφ ) (cid:105) β,µ,n denotes the l.h.s. in (158) there,with κ = − β → − b and µ → nb ) which relates the moments in the circular ensemble to those on the interval, i.ethe moments (cid:104) y k (cid:105) β,a,b,n already used in the previous section. The different specialisations of the parameters leads toother rational functions ˜ M k ( t = − bn, b ) . From (107) and (91-92) in [20] we obtain ˜ M ( t, b ) = − tb b , ˜ M ( t, b ) = bt (cid:0) b + 2 b t − b t + 3 b + 2 bt + 1 (cid:1) ( b + 1) ( b + 2) (2 b + 1) (79)These expressions are again duality invariant. From the FDC we obtain E (cos( kθ m ) e − π √ β tδ N m ) = ˜ M k ( t, E ( e − π √ β tδ N m ) (80)where θ m is the position of the maximum of δ N ( θ ) on the full circle. We can check that for t = 0 all E (cos( kθ m )) = 0 , θ m has a uniform distribution on the circle. For k = 1 we obtain E (cos( θ m ) δ ˜ N m ) = 12 12 π (cid:112) β/ (81)which shows that higher values of the maximum correlate with cos( θ m ) > , i.e. θ m being closer to than to ± π , thepoint at which its value is pinned to zero. Again we see that all connected correlations E c (cos( θ m )( δ ˜ N m ) p ) vanish.For k = 2 we obtain E (cos(2 θ m ) e − π √ β tδ ˜ N m ) = 118 (5 − t ) t ( t + 1) E ( e − π √ β tδ ˜ N m ) (82)which, upon expanding in t yields all joint moments of the form E (cos(2 θ m )( δ ˜ N m )) Consider now the average of cos kφ with respect to the Gibbs measure associated to Z b , defined in (14), on the fullcircle, 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 theeigenvalues E [ (cid:104) cos kφ (cid:105) Z nb ] E [ Z nb ] = (cid:104) cos( kφ ) (cid:105) β,µ,n = ( − k (cid:104) y k (cid:105) β,a,b,n | ( β,a,b ) → ( b, − − b , nb ) = ˜ M k ( t = − bn, b ) (78)The numerator in the l.h.s. of (78) equals Eq. (20) of the text with cos kφ inserted in the integrand. The secondequality is the conjecture obtained in Eqs. (157-158) in [20] (where (cid:104) cos( kφ ) (cid:105) β,µ,n denotes the l.h.s. in (158) there,with κ = − β → − b and µ → nb ) which relates the moments in the circular ensemble to those on the interval, i.ethe moments (cid:104) y k (cid:105) β,a,b,n already used in the previous section. The different specialisations of the parameters leads toother rational functions ˜ M k ( t = − bn, b ) . From (107) and (91-92) in [20] we obtain ˜ M ( t, b ) = − tb b , ˜ M ( t, b ) = bt (cid:0) b + 2 b t − b t + 3 b + 2 bt + 1 (cid:1) ( b + 1) ( b + 2) (2 b + 1) (79)These expressions are again duality invariant. From the FDC we obtain E (cos( kθ m ) e − π √ β tδ N m ) = ˜ M k ( t, E ( e − π √ β tδ N m ) (80)where θ m is the position of the maximum of δ N ( θ ) on the full circle. We can check that for t = 0 all E (cos( kθ m )) = 0 , θ m has a uniform distribution on the circle. For k = 1 we obtain E (cos( θ m ) δ ˜ N m ) = 12 12 π (cid:112) β/ (81)which shows that higher values of the maximum correlate with cos( θ m ) > , i.e. θ m being closer to than to ± π , thepoint at which its value is pinned to zero. Again we see that all connected correlations E c (cos( θ m )( δ ˜ N m ) p ) vanish.For k = 2 we obtain E (cos(2 θ m ) e − π √ β tδ ˜ N m ) = 118 (5 − t ) t ( t + 1) E ( e − π √ β tδ ˜ N m ) (82)which, upon expanding in t yields all joint moments of the form E (cos(2 θ m )( δ ˜ N m )) p ))