arXiv:hep-th/9304011 v1 Apr 5 1993

Therefore, in order to define the double scaling limit we must study the N → ∞ asymptotic
behavior of the kernel K N . As explained in sec. 6.2, matrix model correlation functions
have an asymptotic expansion in 1/N, and are obtained from the asymptotic expansion:
K N ∼ ∑ j≥0
N 1−j K j (λ 1 , λ 2 ) (9.9)
The functions K j have support on an interval 35 I which is independent of j. As a special
case, note in particular that the diagonal of this kernel is the eigenvalue density:
ρ(λ) = K N (λ, λ) . (9.10)
By (9.10) we may identify the interval I with support of the eigenvalue density in perturbation
theory.
Exercise. Eigenvalue Density
Show that ρ(λ) is the probability for finding an eigenvalue with value λ in a random
matrix ensemble described by V (λ). That is, show that it is the matrix expectation value
of
1
N
N∑
δ(λ − λ i ) . (9.11)
i=1
The easiest way to prove our assertions about the nature of the eigenvalue densities
proceeds by studying the correlation functions of the resolvent operators Ŵ (ζ). Note that
Ŵ (ζ) is only defined for ζ off the real axis since φ has real eigenvalues. Moreover, the
discontinuity of Ŵ (ζ) across the real axis is equal to the eigenvalue density.
Solving the quadratic equation we see that the roots of the polynomials define several
branch points for 〈 Ŵ (ζ) 〉 , and since ρ(λ) is the discontinuity of 〈 Ŵ (ζ) 〉 , the
h=0 h=0
support of the genus zero eigenvalue density must lie on an interval or finite union of
intervals.
35 In more complicated cases the support can be on unions of intervals.
107