Spectrum of multispace Euclidean random matrices

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.

Figure 1

Basic statistical ensemble of MERM considered in this work. $L=3$ sets of $N=5$ points, ${\mathbf{r}}_i^{\ell }$, with $\ell =1,\cdots ,L$ and $i=1,\cdots ,N$ are drawn uniformly at random in unit squares ${\mathcal{H}}_2$ (dimension $D=2)$. Points are represented by crosses, whose colors identify their indices $i$. The MERM is defined through (2), where $\mathrm{\Gamma }$ is a generic function of the distance between points, see main text. A possible choice for $\mathrm{\Gamma }$, inspired from the so-called place cells in neuroscience, is the overlap (common area) between pairs of disks of surface ${\varphi }_0<1$ and having centers ${\mathbf{r}}_i^{\ell }$ in each space $\ell$.

Figure 2

$\left(\mathrm{a}\right)z$ vs. $s$, see (19), close to the origin $\left(s=0\right)$, for different values of $\alpha$. $\left(\mathrm{b}\right)$ Support of the spectrum for different values of $\alpha$: black segments show the interval of eigenvalues $z$ with nonzero density $\rho \left(z\right)$. Results obtained by taking for $\mathrm{\Gamma }$ the overlap (common length) between segments of length ${\varphi }_0=.2$, centered in points ${\mathbf{r}}_i^{\ell }$ randomly drawn in the unit interval ${\mathcal{H}}_1\left(D=1\right)$, more precisely $\mathrm{\Gamma }\left(\right|{\mathbf{r}}_i^{\ell }-{\mathbf{r}}_j^{\ell }\left|\right)={\varphi }_0-|{\mathbf{r}}_i^{\ell }-{\mathbf{r}}_j^{\ell }|$.

Figure 3

Density of eigenvalues of $\mathbf{C}$, without the extensive eigenvalue $z_{\mathrm{ext}}$, for various values of $\alpha$. Orange: Results from (38). Blue: Outputs of numerical diagonalization for $N=2500$. Same model as in Fig. 2.

Figure 4

(a) Histogram of the coefficients ${\gamma }_{\ell }$ and ${\delta }_{\ell }$ for different values of $N$. Results correspond to the $k=1$ connected component of eigenvalues in dimension $D=1$ and for $L=5$ spaces, averaged over 50 samples. Same model as in Fig. 2. (b) Histograms of the projections of eigenvectors $\mathbf{v}$ to the $k=2$ Fourier modes of the ERMs. (c) Histograms of the norm of the orthogonal component $\mathbf{R}$, see (46).

Figure 5

Top panels: Sketches of the model variants, respectively (a) dilution, (b) spheres of different sizes and (c) multiple spheres. Bottom panels: (d) Density of eigenvalues for the MERM for ${\varphi }_0=0.2$ with different dilution fractions ${\rho }_0$. $\left(\mathrm{e}\right)$ Density of eigenvalues of MERM with different fractions ${\rho }_{{\varphi }_0=0.2}$ of spheres with volume ${\varphi }_0=0.2$ and ${\rho }_{{\varphi }_0=0.4}$ of spheres with volume ${\varphi }_0=0.4$ in each space. $\left(\mathrm{f}\right)$ Density of eigenvalues of MERM for ${\varphi }_0=0.2$ with $c=2$ spheres for each index $i$ in each map. Parameters: $N=2500,D=1,\alpha =1$. In all cases we do not show the extensive eigenvalue.