10.5 Bounds for the domain of non-vanishing eigenvalue density
10.5.3 Bounds for all t
Our subsequent work confirms the findings in Ref. [3], which, in turn, imply the existence of an annulus with a γ(t) obeying our considerations. For N = ∞, we will show below (see Eq. (11.45)) that the inverse function toγ(t), which we callT(γ) withγ >0, is given by
T(γ) = 2γtanhγ
2, T(γ(t)) =t . (10.65)
The previously presented asymptotic results are recovered since
T(γ)≈γ2 for γ →0, (10.66)
T(γ)≈2γ for γ → ∞. (10.67)
Figure 33 shows a plot ofγ(t) together with the approximations for small and large t.
The approximations in the two regimest→0 andt→ ∞ differ in the order inεone goes to. With either choice, one obtains a finite expression if t is finite, so the truncation of the expansion inεis self-consistent. When the full limitn→ ∞,ε→0 is studied at fixed arbitrary t, going to orderε2 should reproduce both asymptotic results in t, and we shall see that this indeed happens. Note that the crossover between the two asymptotic regimes occurs roughly where √
t =t/2, which means t= 4. It will turn out that as t increases, the spectrum encircles the origin first at a critical value of t= 4 (as in the unitary case).
In some rough sense, this is the point where the lack of commutativity among the factors in the product of matrices becomes qualitatively important.
18The O(n) rotation matrix is given by the matrix S introduced in App. B.3 (with the replacement N →n).
148 10.5 Bounds for the domain of non-vanishing eigenvalue density
2 4 6 8 10
t 1
2 3 4 5 Γ
Figure 33: The red curve showsγ(t) obtained by solving Eq. (10.65) numerically, the blue (dashed) curve shows the large-tapproximation (10.59), the green (dotted) curve corresponds to the small-t approximation (10.64).
11 Saddle-point analysis for the basic model
11.1 Average of products of characteristic polynomials
It is difficult to derive a closed formula for the distribution of all the matrix entries of the product matrix W for generalN. We are interested in just the spectral properties of W, however, even this is difficult to obtain for arbitrary finite N. The main complication in the analysis of the multiplicative random complex matrix model is due to the fact that the matrix product is non-commutative and that the eigenvalues populate two-dimensional domains in the complex plane, rather than one-dimensional line segments.
The natural definition of the spectral (surface) density in the complex plane is given by
ρ(z) = 1 N
* N X
i=1
δ(z−zi(W)) +
, (11.1)
where the eigenvalues of a fixed matrix W are denoted by zi(W),i= 1, . . . , N, andh. . .i denotes the average over the matricesCj definingW. Since a representation of the complex delta function is given by
δ(z) = 1 π lim
ε→0
ε2
(|z|2+ε2)2 , (11.2)
the surface eigenvalue density can be obtained from (cf. Ref. [3]) Fε(z, z∗) = 1
N Tr logh
(z−W)(z∗−W†) +ε2i
= 1
N log deth
(z−W)(z∗−W†) +ε2i
. (11.3)
Due to
∂2
∂z∂z∗Fε(z, z∗) = 1
N Tr ε2
[(z−W)(z∗−W†) +ε2]2
!
= 1 N
N
X
i=1
ε2
(|z−zi(W)|2+ε2)2 , (11.4) the eigenvalue density is given by
ρ(z) = 1 π
D
ε→0lim∂z∂z∗Fε(z, z∗)E
. (11.5)
Rather than calculating ρ(z) directly, we will focus on the averages of characteristic polynomials related to W in the following sections. In the unitary case, we have seen that partial information about the distribution of eigenvalues of the unitary matrix W can be obtained from hdet(z−W)i. (We have found that the locations of the zeros of the average characteristic polynomial provide good approximations for the peaks of the true eigenvalue distribution on the unit circle and that this approximation becomes more and more accurate with increasing N, cf. Sec. 9.)
In analogy to the unitary model, we expect that information about the surface eigen-value density in the multiplicative random complex matrix model can be extracted from averages of characteristic polynomials, too. These polynomials are generating function-als for various moments of the eigenvalue distribution. It turns out that the calculation of some simple characteristic polynomials is feasible. Below, we will first show that the
150 11.1 Average of products of characteristic polynomials
results obtained in Ref. [3] for the basic random matrix model can be reproduced with the help of averages of characteristic polynomials. Then, we will carry over this approach to the generalized model, which allows for a smooth interpolation between the unitary and the basic complex model. In both cases, our findings are confirmed with extensive numerical simulations.
In contrast to the unitary case, hdet(z−W)i carries no information in the complex model: Due to the invariance ofP(C) underC →CeiΦ, we obtain fork∈N0
D Cabk
E
= Z
dµ(C)P(C)Cabk = N π
Z
dµ(Cab)e−N|Cab|2Cabk
= N π
Z 2π 0
dφ Z ∞
0
rdre−N r2rkeiφk =δk0, (11.6) i.e., averages of products of an arbitrary number of factors Cj vanish (as long as there is no contribution from a Cj†). Hence, expanding the matrix elements Wab in powers of matrix elements of the factorsCj leads to
h(Wn)abi=δab. (11.7)
In averages involving only Wn (or Wn†, bot not both), we can therefore make the re-placement Wn → 1. The average of the characteristic polynomial, e.g., is simply given by
hdet(z−Wn)i= (z−1)N. (11.8) The first non-trivial case is
Q(z, z∗) =
|det(z−Wn)|2
, (11.9)
and from now on we focus on the calculation of the above observable in the limitn→ ∞, ε→0 with t=ε2n held fixed (sometimes referred to as “the limit” in the following).
If one applies large-N factorization to
|det(z−Wn)|2
(i.e., assuming that the av-erage of the product can be replaced by the product of the avav-erages, see Sec. 3.6.2) one gets holomorphic factorization,
|det(z−Wn)|2
=|z−1|2N, and all eigenvalues seem to have to be unity. For any t > 0, holomorphic factorization will hold for z-values close to 0 and z-values close to ∞. These two regions are outside the annulus defined by γ(t) (cf. Eq. (10.26)). We will observe below that the full holomorphic factorized regime pene-trates the annulus and is connected for sufficiently smalltbut splits into two disconnected components for t larger than a critical value. There are two disconnected components when the eigenvalue support, always contained within the annulus defined by γ(t), fully encircles the originz= 0.
We wish to calculate Q(z, z∗) as a function of t and confirm that at infinite N the transition we are looking for indeed occurs. As a first step, we need to find a way to disentangle the non-Abelian product defining W. Then, we can make the N-dependence explicit by integrating out all degrees of freedom whose multiplicity isN-dependent. This allows us to take N → ∞ which, as usual, becomes a saddle-point problem. In the following, we analyze the saddle-point problem partially, only to the point where we can identify the transition we are interested in.
11.2 Grassmann-integral representation of characteristic polynomials