## Diffusion and localization in sparse networks

### Real symmetric matrices

Our interest is in real symmetric conservative matrices. $\mathbf{W} = \left(\matrix{ -\gamma_{1} & w_{12} & w_{13} & \cdots & \cdots \cr w_{12} & -\gamma_2 & w_{23} & \cr w_{13} & w_{23} & -\gamma_3 & \cr \vdots & & & \ddots\cr \vdots & & & & \ddots}\right)$ The "conservative" means that each row's sum is zero, or: $\gamma_n \ \ \equiv \ \ \sum_{m\neq n} w_{mn}$

### Motivation

Such matrices appear in rate equations: $\frac{d}{dt}{\mathbf p} \ \ = \ \ \mathbf{W}\cdot\mathbf{p}$ in Newtons II law for systems of masses and springs, which are used as heat transport models: $\frac{d^2}{dt^2}{\mathbf q} \ \ = \ \ -\mathbf{K}\cdot\mathbf{q}$ and in Schrödinger's equation $\frac{d}{dt}{\mathbf \psi} \ \ = \ \ -i {\mathcal{H}}\cdot\mathbf{\psi}$

### Sparsity

We focus on sparse/glassy networks, where by sparse we mean that a small number of elements is larger by orders of magnitude than the rest.

As two examples we use:

2. #### Mott random site model

In this model, the sites are randomly distributed in space, and the rates depend on the distance between sites: $w_{nm} \ \ =\ \ w_0 \cdot \exp\left(-\frac{\left|r_n-r_m\right|}{\xi} - \epsilon_{nm}\right)$ Where $$\xi$$ and $$w_0$$ are system parameters (defining the time and space units), and $$\epsilon$$ a bond specific parameter. In the degenerate version, $$\epsilon=0$$ for all the bonds, while in the non-degenerate version $$\epsilon_{nm} = \textrm{uniform}[0,\infty]$$.

The dimensionless parameter defining sparsity in this model is $s \ \ =\ \ \frac{\xi}{r_0}$ where $$r_0$$ is the typical distance between sites.

3. ### Determining the diffusion coefficient $$D$$

The long term behavior of the system is characterized by a diffusion coefficient $$D$$. It is defined by the long term spreading: $S(t)\quad = \quad \left\langle r^2(t)\right\rangle \quad \sim \quad D t$ And it is also related to the long time survival probability: $\mathcal{P}(t)\quad \sim \quad \frac{1}{(Dt)^{1/2}}$ And to the low eigenvalue distribution (by Laplace transform): $\mathcal{N}(\lambda) \quad \sim \quad \left[ \frac{\lambda}{D}\right]^{d/2}$

### Effective range hopping

For a dense system, (large $$\xi$$ compared to typical site distance), the diffusion coefficient can be estimated by a linear equation (linear in the rates): $D_{\textrm{linear}} = \frac{1}{2d}\sum_r w(r) r^2$ We present the ERH procedure to estimate $$D$$, based on resistor network analysis, with a smooth cross-over between the dense and sparse regimes.

#### Results for the degenerate model

$D \quad =\quad \textrm{EXP}_{d+2}\left(\frac{1}{s_\textrm{eff}}\right) \mbox{e}^{-1/s_\textrm{eff}} D_{\textrm{linear}} \\ s_\textrm{eff}\quad=\quad \left(\frac{d}{\Omega_d} n_c\right)^{-1/d}\frac{\xi}{r_0} \\ \textrm{EXP}_{l}(x) \quad = \quad \sum_{k=0}^l \frac{1}{k!}x^k$

• The parameter $$n_c$$ is the average number of bonds required to get percolation. For $$d=1$$ $$n_c=2$$ and for $$d=2$$ $$n_c\approx 4.5$$, based on studies of disk-percolation.
• If we disregard the percolative nature (by setting $$n_c=0$$), we get the linear estimate back.
• In the limit $$s_\textrm{eff}\ll 1$$ we obtain VRH-like behavior: $D\sim e^{-1/s_\textrm{eff}}$

#### Results for the non-degenerate "Mott" model

In this case, $$s_\textrm{eff}$$ depends explicitly on the temperature, i.e.: $s_{\textrm{eff}}\quad=\quad \left(\frac{d}{\Omega_d} n_c \left(\frac{T}{\Delta_\xi}\right)\right)^{-1/(d+1)}$ Where $$\Delta_\xi$$ is the mean level spacing. Apart from that, the only change in $$D$$ is the polynomial order: $D \quad =\quad \textrm{EXP}_{\color{red}{d+3}}\left(\frac{1}{s_{\textrm{eff}}}\right) \mbox{e}^{-1/s_{\textrm{eff}}} D_{linear}$ In the limit $$s\ll 1$$, we get the familiar VRH estimate that has been presented long ago by Nevill Francis Mott. $D \sim \left(\frac{1}{T}\right)^{2/(d+1)} \exp\left[-\left(\frac{T_0}{T}\right)^{1/(d+1)}\right]$

Diffusion in sparse networks: linear to semi-linear crossover [arXiv] [pdf],
Y. de Leeuw, D. Cohen, Phys. Rev. E 86, 051120 (2012).