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Register a new userOn the Spectrum of Multi-Space Euclidean Random Matrices
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We consider the additive superimposition of an extensive number of independent Euclidean Random Matrices in the high-density regime. The resolvent is computed with techniques from free probability theory, as well as with the replica method of statistical physics of disordered systems. Results for the spectrum and eigenmodes are shown for a few applications relevant to computational neuroscience, and are corroborated by numerical simulations.
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We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected Levy matrices. We derive a critical line separating localized from extended states in the case of Levy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.
By calculating all terms of the high-density expansion of the euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system we show that the low-frequency behavior of the self energy is given by $Sigma(k,z)propto k^2z^{d/2}$ and not $Sigma(k,z)propto k^2z^{(d-2)/2}$, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form $Z(t)propto t^{(d+2)/2}$.
We solve the q-state Potts model with anti-ferromagnetic interactions on large random lattices of finite coordination. Due to the frustration induced by the large loops and to the local tree-like structure of the lattice this model behaves as a mean field spin glass. We use the cavity method to compute the temperature-coordination phase diagram and to determine the location of the dynamic and static glass transitions, and of the Gardner instability. We show that for q>=4 the model possesses a phenomenology similar to the one observed in structural glasses. We also illustrate the links between the positive and the zero-temperature cavity approaches, and discuss the consequences for the coloring of random graphs. In particular we argue that in the colorable region the one-step replica symmetry breaking solution is stable towards more steps of replica symmetry breaking.
The phase diagram of the random field Ising model on the Bethe lattice with a symmetric dichotomous random field is closely investigated with respect to the transition between the ferromagnetic and paramagnetic regime. Refining arguments of Bleher, Ruiz and Zagrebnov [J. Stat. Phys. 93, 33 (1998)] an exact upper bound for the existence of a unique paramagnetic phase is found which considerably improves the earlier results. Several numerical estimates of transition lines between a ferromagnetic and a paramagnetic regime are presented. The obtained results do not coincide with a lower bound for the onset of ferromagnetism proposed by Bruinsma [Phys. Rev. B 30, 289 (1984)]. If the latter one proves correct this would hint to a region of coexistence of stable ferromagnetic phases and a stable paramagnetic phase.
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.
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