Frequency spacing probes vibrational coupling
M. Mumpower, E. Bettelheim, TW
Count number of frequency spacings    xi ¼ wi+1 - wi 
Simplest distribution: omegas are random
Vibrations are decoupled
Number of pairs
P(x)
x
e-x
Spacings of lowest jammed modes
Number
w
Fits expected formula for coupled modes
(Wigner-Dyson random matrix)
x
P(x)
Thus, the anomalous modes are coupled.