lecture - TU Chemnitz by ewghwehws

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									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
The magnetic adjacency spectral statistics

								
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