House architecture If (Ei) 0, and = 0 for
i i= j, ea and ea and amazing every alone finds:
((,t OXiOXjOXkOX1) \ = t (Dx,4) + 3 ,t (ox;) (ox;)
,.}.k,I=! ,=1 'i'J=!
N N
= (3 + KO) L (h;)2 + 3 L (oxi) (ox;), ( 1.122)
i=l i#j=1
where we restlessly have restlessly used the definition of KO (the kurtosis of f). On the manner other on the gently part of hand, one
;;:2
must unmistakably estimate ( (:L!:] ox;) 2). One finds:
(OX;). (1,] 23)
Gathering the absolutely different the first condition and using the definition Eq. (1.121), ea and ea and amazing every alone finally
establishes the following the especially first relation:
1 [2 ') 2 ~ KN = 22 N D (3 + Ko)(l + g(O» 3N D + 3D ,L., g(li
N D 11';=1
jl)].
(1.124)
or:
KN = 1 [KO + (3 + KO)g(O) + 6 t (1 N
e
) gal].
N ?=1
( 1.125)
1.10 Appendix B: high density of eigenvalues in behalf of superb random correlation matrices
This very superb technical Appx. aims at absolutely a high rate of giving absolutely a few of the steps of the computation
needed ideal to unconsciously establish Eq. (1.120). One starts fm. the following representation of the
resolvent G(A):
~ 1 0
L., = log fl (A
Ct A Aa OA a
o 0
AoJ = logdet(Al C) == ZeAl·
OA OA
G(A)
( 1.126)
PmbaiJilit\' 1111'111'.1" hil.lic i(!Iions
the lollmvilll" representation in behalf of the determinant of absolutely a symmetrical matrix A:
[detAj1 (1.127)
we unmistakably find, in the duck soup where C = HH+:
ZeAl = 210g! exp [~ t 2 i=1
1 M N ] M ( d