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

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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 Tₙ. In the localized phase, its logarithm follows the traveling wave form Tₙ ≃ Tₙ + t* where (i) the disorder-averaged value moves linearly (Tₙ) ≃ - Nξloc and the localization length diverges as ξloc (W-Wc)-νloc with νloc=1 (ii) the variable t* is a fixed random variable with a power-law tail P*(t*) 1/(t*)¹⁺β⁽ʷ⁾ for large t* with 0<β(W) ≤ 1/2, so that all integer moments of Tₙ are governed by rare events. In the delocalized phase, the transmission Tₙ remains a finite random variable as N → ∞, and we measure near criticality the essential singularity (T) - | Wc-W |⁻κᵗ with κₜ 0.25. We then consider the statistical properties of normalized eigenstates, in particular the entropy and the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical entropy diverges as (W-Wc)⁻ νˢ with νₛ 1.5, whereas it grows linearly in N in the delocalized phase. Finally for the I.P.R., 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 towards the existence of several exponents ν at criticality.