X
p,q=0
1
p+q+ 1upvqe−2tC(p,q,N)ˆ MA(p, N)MS(q, N). (8.36) We observe that
C(p, q,ˆ −N) = ˆC(q, p, N), (8.37a) MA(p,−N) = (−1)p+1MS(p, N), (8.37b) MS(q,−N) = (−1)q+1MA(q, N). (8.37c) The entire dependence on N in Eq. (8.36) is explicit, and the function ¯R(u, v, N) remains well-defined for N < 0, so long as the fixed parameter t is positive. With N > 0, this leads to
R(u, v, N) = 1 +¯ −u−v
−N
∞
X
p,q=0
(−u)p(−v)q
p+q+ 1 MS(p,−N)MA(q,−N)e−t2C(q,p,−N)ˆ . (8.38) Interchanging the dummy summation labelsp and q we get
R(u, v, N) = ¯¯ R(−v,−u,−N). (8.39) Writing
R(u, v, N) = 1 +¯ u+v
N Ω(u, v, N) (8.40)
results in
Ω(u, v, N) = Ω(−v,−u,−N). (8.41)
Now set u=−v. Ω(−v, v, N) is finite fort >0, and we have
Ω(−v, v, N) = Ω(−v, v,−N). (8.42)
Ω(−v, v, N) determinesρtrueN (θ, t) via Eq. (8.23) because of Ω(−v, v, N) =−N2GtrueN,−(v, t), i.e.,
ρtrueN (θ, t) = 1 + 2 N2 lim
ε→0+Re [vΩ(−v, v, N)], v=eiθ−ε. (8.43) At this point we realize that we have definedρtrueN (θ, t) for negative integerN, too:
ρtrue−N(θ, t) = 1 + 2 N2 lim
ε→0+Re [vΩ(−v, v,−N)] =ρtrueN (θ, t), (8.44) where in the last step we have made use of Eq. (8.42).
8.7 Large-N asymptotics
If one could expand ρtrueN (θ, t) in N around N = 0, only even powers of N would enter.
However, all one can do is an asymptotic expansion in 1/N, and then odd powers can appear (one can think of the asymptotic expansion as an expansion in 1/|N|).
124 8.7 Large-N asymptotics
We now turn to the integral representation to take the first steps in a 1/N expansion of ρtrueN (θ, t). Shifting integration variables x → x+ (t/2) 1/N+ 2/N2
and y → y− i(t/2) 1/N −2/N2
in Eq. (8.32), we obtain GtrueN,−(v, t) =−N Since this integral representation was derived for |v| <1, we set v =eiθ−ε with |θ| ≤π, ε >0, and take the limit ε→0+ at the end. We write Eq. (8.45) as At large N, the integrals over x and y decouple at leading order and can be done in-dependently by saddle-point approximations. Let us start with the integral over y since it is conceptually simpler. The y-dependent coefficient of the term in the exponent in Eq. (8.46) that is proportional to−N is given by
f¯(y) = 1 exactly the same integrand that was already considered in Sec. 7, with the replacements T →tand z→1/vρ(with|vρ|<1) and with an integration over uthat is now along the line from−∞+it/2 to +∞+it/2. Since there are no singularities between this line and the real-uaxis, we can change the integration path to be along the real-u(or imaginary-U) axis. Now everything goes through as in Sec. 7. The saddle-point equation reads
e−tUU + 1/2 U −1/2 = 1
vρ, (8.48)
which is equivalent to Eq. (7.3). In Fig. 27 we show the contours in the complex-U plane on which the solutions of the saddle-point equation have to lie. In analogy to Eq. (8.49), those contours are now determined by
Ui2 =Urcoth(T Ur+ε−logρ)−Ur2−1
4. (8.49)
(For sufficiently smallρ, we now encounter the case mentioned in Sec. 7.1 where the closed contour in the left half-plane is missing for t > 4.) The relevant saddle point, which we denote byy0(θ, t, ρ), is again on the closed contour in the right half-plane. For decreasing ρ this contour contracts, but this makes no difference to our analysis. The result for the y-integral is given by an expression similar to Eq. (7.8).
We now turn to the integral over x. The x-dependent coefficient of the term in the exponent in Eq. (8.46) that is proportional to−N is given by
f(x) =˜ 1 considered in Sec. 7 and the saddle-point equation (8.48), except that the integration is now along the real-U axis. The positions of the saddle points of thex-integral are obtained
-1.0 -0.5 0.0 0.5 1.0 -0.5
0.0 0.5
-1.0 -0.5 0.0 0.5 1.0 -0.5
0.0 0.5
-1.0 -0.5 0.0 0.5 1.0 -0.5
0.0 0.5
Figure 27: Contours of solutions of Eq. (8.49) in the complex-U plane att= 3 (top, left),t= 4 (top, right), and t = 5 (left) for ρ = 1 (black), ρ= 0.9 (red),ρ= 0.6 (green), andρ= 0.3 (blue).
In the figures (but not in the analysis) we have taken|v|= 1 for simplicity.
by rotating the saddles of they-integral by−π/2 in the complex-U plane, i.e., xs=−iys. At a saddle point, we have
f˜00(xs) = 1 t +xs
t
1 +xs t
= ¯f00(ys), (8.51) and therefore the directions of steepest descent through a saddleysand the corresponding saddlexs=−iys are identical (no rotation). By analyzing the directions along which the phase of the integrand is constant, we find that the integration contour can always be deformed to go through the (single) saddle-point in the right half-plane in the direction of steepest descent. Depending on the parameters ρ, v, and t, there is either one or no additional saddle point on the contour(s) in the left half-plane through which we can also go in the direction of steepest descent. If there is such an additional saddle point, we find that its contribution to the integral is always exponentially suppressed in N compared to the saddle point in the right half-plane and can therefore be dropped from the saddle-point analysis. In addition, there are infinitely many more saddle points on the open contour in the left half-plane. However, we cannot deform the integration path to go through these points in the direction of steepest descent and therefore do not need to include them. An example for the location of the saddle points and the deformation of the integration path is given in Fig. 28. To summarize, thex-integral can be approximated by the contribution of the single saddle point in the right half-plane, which again leads to an expression similar to Eq. (7.8).
Combining the saddle-point approximations for the integrals over x and y, we find that, up to exponentially small corrections in N, the integral in Eq. (8.46) is given by
GtrueN,−(v, t) =−N t e−t/2
Z 1
0
dρ 1 2π
2π Nf˜00(x0)
1
(1−vρe−x0−t/2)2 e−x0, (8.52) where x0 =x0(θ, t, ρ) is the dominating saddle point of the x-integral. x0 is a solution of
126 8.7 Large-N asymptotics
-1.0 -0.5 0.0 0.5 1.0 -0.6
-0.4 -0.2 0.0 0.2 0.4 0.6
Figure 28: Example for the location of the saddle points and the deformation of the integration path in the complex-U plane for t = 5 andρ= 0.95. The dashed black curves (two closed, one open) are the curves on which all saddle points have to lie, cf. (7.5). In this example θ = 3.0.
On each of the closed curves there is one saddle point (red dot and blue dot), and on the open curve there are infinitely many saddle points but only one of them in the region shown in the plot (green dot). The thin solid lines are lines of constant Re ˜f(x) and Re ¯f(y). The arrows point in the direction of increasing Re ˜f(x) or decreasing Re ¯f(y). The dashed blue curve is the integration path for the y-integral along the direction of steepest descent. The solid red-blue curve is the integration path for thex-integral along the direction of steepest descent.
the saddle-point equation obtained by differentiating ˜f(x), which can be written as vρe−x0−t/2= x0
x0+t (8.53)
and leads to
1−vρe−x0−t22
= t
t+x0 2
. (8.54)
With Eq. (8.51) we obtain f˜00(x0)
1−vρe−x0−t2 2
= t+x0(t+x0)
(t+x0)2 (8.55)
and
GtrueN,−(v, t) =−1 te−t2
Z 1 0
dρ (t+x0)2
t+x0(t+x0)e−x0. (8.56) Differentiating Eq. (8.53) with respect toρ leads to
∂x0
∂ρ = 1 ρ
x0(t+x0)
t+x0(t+x0) =ve−x0−t/2 (t+x0)2
t+x0(t+x0), (8.57) which yields
GtrueN,−(v, t) =−1 tv
Z 1 0
dρ∂x0
∂ρ =−1
tv[x0(θ, t, ρ= 1)−x0(θ, t, ρ= 0)] . (8.58)
We know from Eq. (8.53) that x0(θ, t, ρ= 0) = 0. Therefore, GtrueN,− is determined by the
Here, we need to keep in mind that we have to pick the solution of Eq. (8.48) which corresponds to the dominating saddle point x0 of the x-integral for |vρ|< 1. Forρ = 1, Eq. (8.48) coincides with the saddle point equation (7.3), which determinesρsym∞ (θ, t), if we replace tby T and 1/v by z, i.e., θ→ −θ (we have always considered|v|<1 and|z|>1;
in the infinite-N limit, we havet =T). By comparison with Eq. (7.12), we observe that to leading order in 1/N
FN,−true(eiθ−ε, t) =eiθ−εGtrueN,−(eiθ−ε, t)−1
2 =−U(θ, t, ρ= 1)
=−FN,+sym(e−iθ+ε, t) =−FN,+sym(eiθ+ε, t)∗ =FN,−sym(eiθ−ε, t), (8.60) where we have used thatU(−θ) =U(θ)∗ and Eq. (5.85), relatingFN,+sym andFN,−sym. Clearly, this implies that the related densities are equivalent in the infinite-N limit, cf. Eqs. (5.23) and (5.86),
Nlim→∞ρtrueN (θ, t) = lim
N→∞ρsymN (θ, t), (8.61)
which in turn is equal to the Durhuus-Olesen resultρ∞(θ, t), cf. Sec. 7.2. To compute the asymptotic expansion of ρtrueN (θ, t) in powers of 1/N (in regions where this makes sense), one has to keep higher orders in the saddle-point approximation (as explained in Sec. 7.3).
8.8 A partial differential equation for the average of the ratio of char-acteristic polynomials at different arguments
In the expression for Ω(u, v, N) that follows from Eq. (8.36), a derivative with respect to t will bring down the Casimir factor from the exponent. Writing
u=−eX+Y , v=eX−Y , fN(X, Y, t) = Ω(u, v, N), (8.62) we can reconstruct the Casimir by derivatives with respect toX and Y. All that enters is the bilinear structure of the Casimir, and we obtain
∂fN
Rescaling X → N X = Z removes all explicit dependence on N in the equation. The equation is linear, so we are free to rescalegN by any power ofN we find convenient. We define