3. Spectral statistics. Random Matrix Theory in a nut shell.

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Presentation transcript:

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 GOE

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

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 d=10

The magnetic adjacency spectral statistics