# lecture - TU Chemnitz by ewghwehws

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• pg 1
```									3. Spectral statistics.
Random Matrix Theory in a nut shell.

Eigenvalues distribution:
Graphs: the spectrum and the spectral statistics
The random G(V,d) ensemble

GUE
GUE
GOE
GOE

1                                                  0.8
0.9
0.7
0.8
0.6
0.7

0.6                                                 0.5

0.5                                                 0.4

0.4
0.3
0.3
0.2
0.2
0.1
0.1

0                                                   0

0   0.5   1   1.5    2    2.5   3   3.5   4         0   0.5   1   1.5    2     2.5   3   3.5   4
s                                                   s
Spectral 2-points correlations:

(mapping the spectrum on the unit circle)

GUE                                           GOE
1.5                                            1.5

1
1

0.5
0.5

0
0
0.5   1   1.5   2     2.5   3   3.5     40   0.5   1    1.5    2     2.5     3    3.5     4
s                                              s
Why do random graphs display the
canonical spectral statistics?

Counting statistics of cycles vs Spectral statistics

The main tool : Trace formulae connecting

spectral information
and
counts of periodic walks on the graph

The periodic walks to be encountered here are special:
Backscattering along the walk is forbidden.
Notation: non-backscattering walks = n.b. walks
Spectral Statistics

Two-point correlation function. However: the
spectral variables are not distributed uniformly and to
compare with RMT they need unfolding
The (not unfolded)
Spectral formfactor

Spectral form factor =
variance of the number of
t-periodic nb - walks

# t-periodic
nb cycles

For t < logV/log (d-1) C_t are distributed as a Poissonian variable
Hence: variance/mean =1 (Bollobas, Wormald, McKay)
Conjecture (assuming RMTfor d-regular graphs):
V=1000          V=1000

d=10