6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble
171
Fig. 6.4 Histogram of the
number of eigenvalues, N(x),
versus eigenvalue, x, for an
N × N , real symmetric
random matrix with matrix
elements distributed
according to GOE and
N = 20,000. A fit to the
Wigner semicircle law gives a
confidence level of 98%,
when using a χ 2 test (plot by
Gursoy Akguc)
The Wigner semicircle law in Eq. (6.74) is most accurate when N →∞. It is
plotted in Fig. 6.2b for the case N = 22, where it may be compared to the
exact results of random matrix theory. We see that, even for this low value of N,
the agreement is quite good. In Fig. 6.4, we show a histogram of the number of
eigenvalues, N(x), versus eigenvalue, x, for an N × N , real symmetric random
matrix with N = 20,000. The matrix elements are distributed according to GOE.
A fit to the Wigner semicircle law (not shown here) gives agreement with a
confidence level of 98% when using a chi-squared test.
We can obtain an expression for the Lagrange multiplier, γ , if we set x = 0 in
Eq. (6.70) and perform the integration. We find γ = Nln[
√
2N ] −
N
2 − N ln[2].
Chi-squared Confidence Test
The chi-squared (χ 2 ) test gives a measure of the confidence with which a histogram can
be fit by a given curve (Brookes and Dick 1969), (Meyer 1975). Let us assume the histogram
contains N data points divided among s bins to form a histogram and that there are n i data
points in the ith bin (i = 1, . . . , s). Let us assume that the theoretical distribution predicts
that the ith bin has ν i data points. Then χ 2 ≡
s
i=1 (ν i − n i ) 2 /ν i . Once the value of χ 2 is
known and any conditions relating the number of data points in the various bins are known
(such as
s
i=1 n i = N ), then tables exist (Brookes and Dick 1969), (Meyer 1975) that allow
one to determine the confidence with which the given theoretical distribution reproduces the
data.
Staircase Function
It is useful to introduce the staircase function, η(x), which is the number of
eigenvalues with value less than x. The staircase function is defined as
η(x) =
x
−∞
dy ρ(y).
(6.75)
The staircase function η W ig (x) obtained from the Wigner semicircle law is
η W ig (x) =
1
π
x
−∞
dy
2N − y 2 =
x
√
2N − x 2
2π
+
N
π
sin
x
√
2N
+
N
2
.
(6.76)
171
Fig. 6.4 Histogram of the
number of eigenvalues, N(x),
versus eigenvalue, x, for an
N × N , real symmetric
random matrix with matrix
elements distributed
according to GOE and
N = 20,000. A fit to the
Wigner semicircle law gives a
confidence level of 98%,
when using a χ 2 test (plot by
Gursoy Akguc)
The Wigner semicircle law in Eq. (6.74) is most accurate when N →∞. It is
plotted in Fig. 6.2b for the case N = 22, where it may be compared to the
exact results of random matrix theory. We see that, even for this low value of N,
the agreement is quite good. In Fig. 6.4, we show a histogram of the number of
eigenvalues, N(x), versus eigenvalue, x, for an N × N , real symmetric random
matrix with N = 20,000. The matrix elements are distributed according to GOE.
A fit to the Wigner semicircle law (not shown here) gives agreement with a
confidence level of 98% when using a chi-squared test.
We can obtain an expression for the Lagrange multiplier, γ , if we set x = 0 in
Eq. (6.70) and perform the integration. We find γ = Nln[
√
2N ] −
N
2 − N ln[2].
Chi-squared Confidence Test
The chi-squared (χ 2 ) test gives a measure of the confidence with which a histogram can
be fit by a given curve (Brookes and Dick 1969), (Meyer 1975). Let us assume the histogram
contains N data points divided among s bins to form a histogram and that there are n i data
points in the ith bin (i = 1, . . . , s). Let us assume that the theoretical distribution predicts
that the ith bin has ν i data points. Then χ 2 ≡
s
i=1 (ν i − n i ) 2 /ν i . Once the value of χ 2 is
known and any conditions relating the number of data points in the various bins are known
(such as
s
i=1 n i = N ), then tables exist (Brookes and Dick 1969), (Meyer 1975) that allow
one to determine the confidence with which the given theoretical distribution reproduces the
data.
Staircase Function
It is useful to introduce the staircase function, η(x), which is the number of
eigenvalues with value less than x. The staircase function is defined as
η(x) =
x
−∞
dy ρ(y).
(6.75)
The staircase function η W ig (x) obtained from the Wigner semicircle law is
η W ig (x) =
1
π
x
−∞
dy
2N − y 2 =
x
√
2N − x 2
2π
+
N
π
sin
x
√
2N
+
N
2
.
(6.76)