arXiv:hep-th/9304011 v1 Apr 5 1993

(The 10 rrr terms start with r n (r 2 n + r 2 n+1 + r 2 n−1 + . . .) and may be found e.g. in [98].)
As mentioned before (7.6), in the large N limit the index n becomes a continuous
variable ξ, and we have r n /N → r(ξ) and r n±1 /N → r(ξ ± ε), where ε ≡ 1/N. To leading
order in 1/N, (7.10) reduces to
gξ = r + 6r 2 + 30br 3 = W (r)
= g c + 1 2 W ′′ | r=rc
(
r(ξ) − rc
) 2
+ . . . .
(7.11)
In the second line, we have expanded W (r) for r near a critical point r c at which W ′ | r=rc =
0 (which always exists without any fine tuning of the parameter b), and g c ≡ W (r c ). We
see from (7.11) that
r − r c ∼ (g c − gξ) 1/2 .
For a general potential V (λ) = 1 ∑
2g p a p λ 2p in (7.9), we would have
W (r) = ∑ p
a p
(2p − 1)!
(p − 1)! 2 rp . (7.12)
To make contact with the 2d gravity ideas of chapt. 6, let us suppose more generally
that the leading singular behavior of f(ξ) ( = r(ξ) ) for large N is
f(ξ) − f c ∼ (g c − gξ) −Γ str
(7.13)
for g near some g c (and ξ near 1). (We shall see that Γ str in the above coincides with the
critical exponent Γ str defined in (6.12).) The behavior of (7.6) for g near g c is then
∫
1
1
N 2 Z ∼
0
dξ (1 − ξ)(g c − gξ) −Γ str
∼ (1 − ξ)(g c − gξ) −Γ str+1
∣ 1 +
0
∼ (g c − g) −Γ str+2 ∼ ∑ n
∫ 1
0
n Γ str−3 (g/g c ) n .
dξ (g c − gξ) −Γ str+1
(7.14)
Comparison with (6.12) shows that the large area (large n) behavior identifies the exponent
Γ str in (7.13) with the critical exponent defined earlier.
derivative of Z with respect to x = g c − g has leading singular behavior
We also note that the second
Z ′′ ∼ (g c − g) −Γ str
∼ f(1) . (7.15)
From (7.13) and (7.14) we see that the behavior in (7.11) implies a critical exponent
Γ str = −1/2. From (6.13), we see that this corresponds to the case D = 0, i.e. to pure
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