Localization properties of the sparse Barrat-Mézard trap model

Inspired by works on the Anderson model on sparse graphs, we devise a method to analyze the localization properties of sparse systems that may be solved using cavity theory. We apply this method to study the properties of the eigenvectors of the master operator of the sparse Barrat-Mézard trap model, with an emphasis on the extended phase. As probes for localization, we consider the inverse participation ratio and the correlation volume, both dependent on the distribution of the diagonal elements of the resolvent. Our results reveal a rich and nontrivial behavior of the estimators across the spectrum of relaxation rates and an interplay between entropic and activation mechanisms of relaxation that give rise to localized modes embedded in the bulk of extended states. We characterize this route to localization and find it to be distinct from the paradigmatic Anderson model or standard random matrix systems.

Figure 1

Inverse participation ratio (multiplied by $N$). Scatter plot: Direct diagonalization [Eq. (13)]. Comparison with estimator Eq. (14), obtained by either single instance (dash-dotted line) or population dynamics (dashed line). Size of the network $N=2^{14}$, population size $N_p={10}^5$ with the same effective $N=2^{14}$. Temperature $T=0.9$ (top panel), $T=0.5$ (bottom panel), and eigenvalue grid size $\mathrm{\Delta }\lambda =0.01$.

Figure 2

Histograms of the full distribution $P\left(\mathrm{Im}G\right)$ (after a logarithmic transformation) for $\lambda =-1/2$. Temperature $T=0.9$ (a), $T=0.5$ (b). Single instance (dash-dotted line) for a network with size $N=2^{14}$. Population dynamics (dashed line) with the same effective $N=2^{14}$ and population size $N_p={10}^3$.

Figure 3

$N$ dependence of the product $N{I}_2$ for two different temperatures. (a) $T=0.9$. (b) $T=0.5$. Eigenvalue grid size $\mathrm{\Delta }\lambda =0.01$. Population size $N_p={10}^4$. Results are averages over one hundred independent runs.

Figure 4

$N$ dependence of the correlation volume $N_{\xi }$ for two different temperatures. (a) $T=0.9$. (b) $T=0.5$. The raw data are the same as those used for Fig. 3.

Figure 5

Correlation volume $N_{\xi }$ for different temperatures in the glassy regime ($T<1$). Average over ten independent runs. Effective network size $N=2^{44}$. Population size $N_p={10}^5$. Eigenvalue grid size $\mathrm{\Delta }\lambda =0.01$.

Figure 6

Dependence of the correlation volume $N_{\xi }$ on the population size $N_p$. $T=0.38$, $\lambda =-0.5$ (a), and $\lambda =-0.1$ (b). Effective network size $N=2^{44}$. The error bars show half of the standard deviation obtained across ten independent runs.

Figure 7

Cumulative local DOS around the node with maximum value up to distance $r$ [$s_r$ from Eq. (20)] for different temperatures and fixed $\lambda =-1/2$. Results are averages over ten finite instances of size $N=2^{18}$, using local values $\mathrm{Im}{G}_{ii}$ computed by SI. For comparison we also provide the estimated values of $r_{\xi }$ (converted from Fig. 5 and marked by stars).

Figure 8

Correlation function of the local DOS [(a), Eq. (17)] and connected correlation function [(b), Eq. (18)] for different temperatures and $\lambda =-1/2$ on a single instance of size $N=2^{19}$. The stars indicate the correlation length $r_{\xi }$ estimated from PD.

Figure 9

Dependence of $N_{\xi }$ (top panel) and $N{I}_2$ (bottom panel) on effective network size $N$, for $T=0.4$ in a small range around $\lambda =-1/3$. Population size $N_p={10}^5$. Eigenvalue grid size $\mathrm{\Delta }\lambda ={10}^{-3}$.

Figure 10

Histograms of the full distribution $P\left(\mathrm{Im}G\right)$ (after a logarithmic transformation) for temperatures $T=0.4$ (a) and $T=0.7$ (b). The dashed line shows the fit of the tail to $P\left(\mathrm{Im}G\right)\propto {\left(\mathrm{Im}G\right)}^{-\alpha }$ with $\alpha =1.8$ (a) and $\alpha =2.46$ (b). Results are from PD with effective $N=2^{40}$ ($\epsilon \sim {10}^{-12})$ and population size $N_p={10}^5$, for values of $\lambda$ near the spectral peak as shown.

Figure 11

Power-law exponent of the right tail $P\left(\mathrm{Im}G\right)\propto \mathrm{Im}{G}^{-\alpha }$ at $\lambda =-1/3$. Exponents were fitted using the maximum likelihood estimator of Ref. [52]; error bars are obtained by comparing estimates from different fitting intervals $(\mathrm{Im}{G}^{\min },\mathrm{Im}{G}^{\max })$. Dashed line: Guide to the eye, interpolating linearly from $\alpha =2$ at $T=1/2$ to $\alpha =3$ at the glass transition temperature $T=1$.

Figure 12

Cumulative distribution of the local trap depths $E$ around chosen network nodes of depth $E_c$ for $\lambda =-1/3$ and $T=0.4$. Dashed line: Typical nodes. Solid line: Nodes with large $\mathrm{Im}{G}_{ii}$, i.e., in the tail of $P\left(\mathrm{Im}G\right)$ (see Fig. 10). Results from population dynamics with effective $N=2^{40}$ and $N_p={10}^5$. Dotted horizontal line: $P(E-E_c\le x)=2/3$, provided for comparison with the height of the observed plateau.

Figure 13

Typical (a) and mean (b) values of $P\left(\mathrm{Im}G\right)$ for $T=0.5$ in the middle of the spectrum ($\lambda =-1/2$), estimated with exact diagonalization (ED) and the single instance (SI) cavity method, as a function of $\epsilon /\delta$, with $\delta =1/\left[\rho \right(\lambda \left)N\right]$ being the mean level spacing. Size of the instance: $N=2^{14}$.

Figure 14

Inverse participation ratio (multiplied by $N$) for three different single instances of the master operator. Size of the network $N=2^{14}$, temperatures $T=10$ (a), $T=1$ (b), and $T=0.5$ (c). Scatter plot: Direct diagonalization [Eq. (13)]. The dashed line indicates estimator (14), and the dash-dotted line estimator (B3), both obtained using the single instance cavity method [Eqs. (2, 3, 4)] with eigenvalue grid size $\mathrm{\Delta }\lambda =0.01$.