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Random Matrix Theory in a nut shell.

3. Spectral statistics. Random Matrix Theory in a nut shell. Eigenvalues distribution:. Graphs: the spectrum and the spectral statistics. GUE. GOE. The random G (V,d) ensemble. Spectral 2-points correlations:. ( mapping the spectrum on the unit circle). Why do random graphs display the

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Random Matrix Theory in a nut shell.

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  1. 3. Spectral statistics. Random Matrix Theory in a nut shell. Eigenvalues distribution:

  2. Graphs: the spectrum and the spectral statistics

  3. GUE GOE The random G(V,d) ensemble

  4. Spectral 2-points correlations: (mapping the spectrum on the unit circle)

  5. 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 isforbidden. Notation: non-backscattering walks = n.b. walks

  6. Spectral Statistics Two-point correlation function. However: the spectral variables are not distributed uniformly and to compare with RMT they need unfolding

  7. The (not unfolded) Spectral formfactor # t-periodic nb cycles Spectral form factor = variance of the number of t-periodic nb - walks For t < logV/log (d-1) C_t are distributed as a Poissonian variable Hence: variance/mean =1 (Bollobas, Wormald, McKay)

  8. Conjecture (assuming RMTfor d-regular graphs):

  9. V=1000 V=1000 d=10

  10. The magnetic adjacency spectral statistics

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