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We study the finite-time dynamics of an initially localized wave-packet in the Anderson model on the random regular graph (RRG). Considering the full probability distribution $Pi(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$, we give evidence that $Pi(x,t)$ spreads sub-diffusively over a range of disorder strengths, wider than a putative non-ergodic phase. We provide a detailed analysis of the propagation of $Pi(x,t)$ in space-time $(x,t)$ domain, identifying four different regimes. These regimes in $(x,t)$ are determined by the position of a wave-front $X_{text{front}}(t)$, which moves sub-diffusively to the most distant sites $X_{text{front}}(t) sim t^{beta}$ with an exponent $beta < 1$. We support our numerical results by a self-consistent semiclassical picture of wavepacket propagation relating the exponent $beta$ with the relaxation rate of the return probability $Pi(0,t) sim e^{-Gamma t^beta}$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.
We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths:
The Anderson transitions in a random magnetic field in three dimensions are investigated numerically. The critical behavior near the transition point is analyzed in detail by means of the transfer matrix method with high accuracy for systems both wit
We study the performance of different message passing algorithms in the two dimensional Edwards Anderson model. We show that the standard Belief Propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a
We present strong numerical evidence for the existence of a localization-delocalization transition in the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchical models are important because of the well-known mapping
A recent interesting paper [Yucesoy et al. Phys. Rev. Lett. 109, 177204 (2012), arXiv:1206:0783] compares the low-temperature phase of the 3D Edwards-Anderson (EA) model to its mean-field counterpart, the Sherrington-Kirkpatrick (SK) model. The autho