In Eq. (6.58) we have setτ = 4. Alternatively, we can also scaleτ with N in the vicinity of the critical point as in Sec. 5.5.3 , according to
τ = 4
1 +√α
3N
. (6.66)
-4 -2 2 4 Α 0.5
1.0 1.5 2.0 2.5 3.0 Μ1
Figure 22: Plots ofµ1(α), obtained by computing the LHS of Eq. (6.68) numerically at finiteN, for N = 10 (green dots), N = 100 (blue dots), and N = 2000 (red dots). The black curve shows the expected result at infiniteN, following from Eq. (6.67). We observe that the convergence to the infinite-N limit becomes slower with increasingα.
Then, the leading asymptotic behavior of the zeros yk∗ is again given by Eq. (6.62) if we replace µk by µk(α), which are defined to be the zeros of
Fα(µ) = Z
du eµ
“
4iu−u4−√αµu2”
(6.67) with µk(α = 0) =µk. Therefore, we obtain for the extremal angleθM(τ) that
N→∞lim N
3
"
3 4√
2
"
π−θM τ = 4 1 +√α
3N
!##43
=µ1(α) (6.68)
with µ1(α= 0)≈0.8221. See Fig. 22 for a plot of µ1(α) in the range −4≤α ≤4.
112
7 Asymptotic expansion of ρ
symN(θ, T )
The aim of this section is to construct an asymptotic expansion ofρsymN (θ, T) in powers of 1/N. To this end, we perform a saddle-point analysis11 of the integral in Eq. (5.82), from which ρsymN (θ, T) can be obtained via Eqs. (5.84) and (5.86). It is sufficient to study only ψN,+sym(z, T) becauseψN,−sym(z, T) can be obtained from Eq. (5.72) onceψsymN,+(z, T) is known.
7.1 Saddle-point analysis
For|z|= 1, the integrand of Eq. (5.82) has singularities on the real-u axis. We therefore set z = eε+iθ, where ε > 0 ensures that |z| > 1 but will later be taken to zero. The integrand of Eq. (5.82) can be written as exp(−N f(u, z)) with
f(u, z) = u2
2T + log
ze−iu2 −eiu2
. (7.1)
We now look for saddle points of the integrand in the complex-u plane, which we label by ¯u =iT U(θ, T), where U(θ, T) = Ur(θ, T) +iUi(θ, T) is a complex-valued function of θ and T (of course U depends also on ε as long as we keep ε >0). In the following, the explicit z-dependence of f(u, z) is often suppressed to simplify the notation, we simply write f(u)≡f(u, z). Due to
f0(u)≡∂uf(u, z) = u T − i
2
ze−iu+ 1
ze−iu−1, (7.2)
the saddle-point equation,f0(iT U) = 0, turns out to be e−T U(θ,T)U(θ, T) + 1/2
U(θ, T)−1/2 =z=eε+iθ. (7.3) For ε = 0, this is Eq. (5.49) in Ref. [49] and is related to the inviscid complex Burgers equation via Eq. (5.44) therein (cf. also Sec. 11.4.3 below). In the present notation, the latter equation has the form
∂U
∂T +iU∂U
∂θ = 0. (7.4)
We will show in Sec. 7.2 that the dominating saddle pointU(θ, T) directly determines the infinite-N limit of the density ρsymN (θ, T). In Sec. 4.4, we have seen that the Durhuus-Olesen result for ρ∞(θ, t) is obtained by solving the inviscid complex Burgers equa-tion (4.72), which is equivalent to Eq. (7.4) (up to factors of i). This already indicates that indeed limN→∞ρsymN =ρ∞ as expected, cf. Sec. 7.2.
Taking the absolute value of Eq. (7.3) leads to the equation (for Ur6= 0) Ui2 =Urcoth(T Ur+ε)−Ur2−1
4. (7.5)
For ε = 0, this equation has been investigated also in Ref. [49], cf. Sec. 11.4.3 below.
Equation (7.5) describes one or more curves in the complex-U plane on which the saddle points have to lie (for a given value ofθ, the saddles are isolated points on these curves).
As long as we keep ε > 0, there are no solutions of Eq. (7.3) with Ur = 0. When we set ε = 0, we can expand the RHS of Eq. (7.5) in Ur and obtain at leading order Ui2= 1/T−1/4 (cf. also Eq. (11.52) below), which admits real solutions forT <4. These are the points where the curves of solutions intersect the imaginary axis (for ε = 0 and
11See App. C for a general description of the saddle-point method.
-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 23: Examples of the contours in the complex-U plane described by Eq. (7.5) forT = 3 (top, left), T = 4 (top, right), and T = 5 (left).
The red dashed curves are for small ε >0, while the solid black curves show the limit ε→0. For our saddle-point analysis we keepε >0.
T < 4). There are no such intersections for T > 4. However, when we set Ur = 0, the absolute value of the LHS of Eq. (7.3) is equal to one, which means that every point on the imaginary axis provides a solution for some value of θ if we set ε= 0 (see Fig. 23 for some plots).
Here, we keepε >0 for the time being. The singularities of the integrand of Eq. (5.82) then all have Ur < 0 (they are located between the imaginary axis and those curves of solutions of Eq. (7.5) which are located in the half-planeUr <0).
In Fig. 23, we show typical examples for the curves described by Eq. (7.5) for T <4, T = 4, andT >4, whereεhas been chosen sufficiently close to zero. The closed contours always enclose the points U = 1/2 or U = −1/2. For T > 4 and larger ε, the closed contour in the left half-plane would be missing, but right now we are not concerned with this since we are only interested in the limit ε→0+.
Clearly, every solutionU(θ, T) of Eq. (7.3) has to fulfill Eq. (7.5) by construction. On the other hand, every solution of Eq. (7.5) leads to a solution of Eq. (7.3) for one unique value of θ with −π < θ ≤ π (for a given solution Ur+iUi of Eq. (7.5), this value of θ is obtained simply by evaluating the LHS of Eq. (7.3)). Analyzing Eq. (7.3) numerically, we find for all values of T that for a given value of θ there is always one (and only one) saddle point on the closed contour in the right half-plane, i.e., with Ur >0. This means that there is a one-to-one mapping from θ (with −π < θ ≤ π) to saddle points on this closed contour, cf. Fig. 24.
Note that we are showing the complex-U plane in Fig. 23, in which the original integra-tion contour corresponds to the imaginary axis. The integraintegra-tion contour can be smoothly deformed to go through the (single) saddle point in the right half-plane along a path of steepest descent. No singularities are crossed since they all have Ur <0. There are also saddle points on the contour(s) in the left half-plane (in fact, there are infinitely many on the open contour), but these need not to be considered. Figure 25 shows an example for the location of the saddle points and the deformation of the integration contour in the
114 7.1 Saddle-point analysis
-3 -2 -1 1 2 3
Θ
-0.10 -0.05 0.05 Ur-12, Ui
Figure 24:Plot of the solutionU(θ, T) of Eq. (7.3) which is located on the closed contour in the right half plane as a function ofθforT = 5 andε= 0. The dashed blue curve shows Ur(θ,5)− 12, the red curve showsUi(θ,5).
complex-u plane.
Once the integration contour has been deformed to go through the saddle point, we can safely take the limitε→0+. Parametrizing the contour in the vicinity of the saddle point byu= ¯u+xeiβ, wherexis the new integration variable corresponding to the fluctuations around the saddle andβ is the angle which the path of steepest descent makes with the real-uaxis, ψN,+sym(eiθ, T) is given, up to exponentially small corrections in N, by
ψN,+sym(eiθ, T) = 1 2N
r N
2πT eN T8 −iN θ2 +iβ Z ∞
−∞
dx e−N g(x), (7.6)
g(x) = 1
2T xeiβ+iT U(θ, T)2
+ log sinhiθ−ixeiβ+T U(θ, T)
2 . (7.7)
We can now expandg(x) inx. The linear order vanishes by construction, the second order gives a Gaussian integral overx, resulting in
ψsymN,+(eiθ, T)≈eN T8 +N T U2(θ,T) 2
e−iθ(1/4−U2(θ, T))N/2
p1−T(1/4−U2(θ, T)) . (7.8) Note that the factor e−iθ cannot be pulled out of the term in square brackets because periodicity inθwould be lost.
There is a potential complication. In principle,g00(0) and therefore the denominator in Eq. (7.8) could be zero, which would mean that the integral overx cannot be performed in Gaussian approximation. ForT > 4, it is straightforward to show that g00(0) is never zero. ForT ≤4, one can use Eq. (7.3) to show that g00(0) = 0 only for the saddle points corresponding to the two anglesθ=±θc(T) at which the transition from zero to non-zero
-4 -2 0 2 4
-4 -2 0 2
4 Figure 25: Example for the location of the saddle points and the deformation of the integration con-tour in the complex-u plane forT = 5 and θ = 3 (with smallε >0). The thin solid lines are lines of constant Ref(u), the arrows point in the direction of increasing Ref(u). On each of the closed orange curves there is one saddle point (red dot and blue dot), and on the open orange curve there are in-finitely many saddle points but only one of them in the region shown in the plot (green dot). The thick blue curve is the integration path along the direction of steepest descent through the relevant saddle point.
ρ∞(θ, T) occurs (see Sec. 5.6). This means that the asymptotic expansion in 1/N diverges for|θ|=θc(T), and that it converges ever more slowly as|θ| →θc from below.
Note that for T < 4 and θc(T) ≤ |θ| ≤ π, the function ρsymN (θ, T) is exponentially suppressed in N. The study of the large-N asymptotic behavior in this region requires more work.