Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions

For Anderson localization on the Cayley tree, we study the statistics of various observables as a function of the disorder strength W and the number N of generations. We first consider the Landauer transmission T-N. In the localized phase, its logarithm follows the traveling wave form ln T-N similar or equal to (ln T-N) over bar + ln t* where (i) the disorder-averaged value moves linearly <(ln(T-N))over bar> similar or equal to -N/xi(loc) and the localization length diverges as xi(loc) similar to (W - W-c)-(nu loc) with nu(loc) = 1 and (ii) the variable t* is a fixed random variable with a power-law tail P*(t*) similar to 1/(t*)(1+beta(W)) for large t* with 0 < beta(W) <= 1/2, so that all integer moments of T-N are governed by rare events. In the delocalized phase, the transmission T-N remains a finite random variable as N -> infinity, and we measure near criticality the essential singularity <(ln(T-infinity))over bar> similar to -vertical bar W-c-W vertical bar(-kappa T) with kappa(T) similar to 0.25. We then consider the statistical properties of normalized eigenstates Sigma(x)vertical bar psi(x)vertical bar(2) = 1, in particular the entropy S = -Sigma(x)vertical bar psi(x)vertical bar(2)ln vertical bar psi(x)vertical bar (2) and the inverse participation ratios (IPR) I-q = Sigma(x) vertical bar psi(x)vertical bar(2q). In the localized phase, the typical entropy diverges as S-typ similar to (W - W-c)(-nu S) with nu(S) similar to 1.5, whereas it grows linearly as S-typ (N) similar to N in the delocalized phase. Finally for the IPR, we explain how closely related variables propagate as traveling waves in the delocalized phase. In conclusion, both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions, and the Anderson localization/ delocalization transition then corresponds to a traveling/non-traveling critical point. Moreover, our results point toward the existence of several length scales that diverge with different exponents nu at criticality.