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 $\ln T_N \simeq \bar{\ln T_N} + \ln t^*$ where (i) the disorder-averaged value moves linearly $\bar{\ln (T_N)} \simeq - \frac{N}{\xi_{loc}}$ and the localization length diverges as $\xi_{loc} \sim (W-W_c)^{-\nu_{loc}}$ with $\nu_{loc}=1$ (ii) the variable $t^*$ is a fixed random variable with a power-law tail $P^*(t^*) \sim 1/(t^*)^{1+\beta(W)}$ for large $t^*$ with $0<\beta(W) \leq 1/2$, 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 $N \to \infty$, and we measure near criticality the essential singularity $\bar{\ln (T)} \sim - | W_c-W |^{-\kappa_T}$ with $\kappa_T \sim 0.25$. 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 $(W-W_c)^{- \nu_S}$ with $\nu_S \sim 1.5$, 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 $\nu$ at criticality.Comment: 28 pages, 21 figures, comments welcom

Quasi-stationary regime of a branching random walk in presence of an absorbing wall

A branching random walk in presence of an absorbing wall moving at a constant velocity $v$ undergoes a phase transition as the velocity $v$ of the wall varies. Below the critical velocity $v_c$, the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time $T$. We study the quasi-stationary regime for $v<v_c$ when $T$ is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time $T$. We then use this construction to show that the properties of the quasi-stationary regime are universal when $v\to v_c$. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.Comment: 2 figures, minor corrections, one reference adde

Random tree growth by vertex splitting

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's $\alpha$-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.Comment: 47 page