Anomalous scaling regime for one-dimensional Mott variable-range hopping

Abstract

We derive an anomalous, sub-diffusive scaling limit for a one-dimensional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this 'blocking' mechanism with one of 'trapping'. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.

Figures (6)

• Figure 1: Simulation of $(X_t)_{t\geq0}$ in the cases $\rho=0.7$ (top row) and $\rho=0.95$ (bottom row), for $\beta=\lambda=0$ and $3\cdot 10^6$ steps. The left column shows the process in physical space, with vertical lines indicating the environment $\{\omega_i:i\in\mathbb Z\}$. The vertical lines in the right column denote the coordinates $\{\mathop{\mathrm{sign}}\nolimits(i)R^{0,0}(\omega_0,\omega_i):i\in\mathbb Z\}$ in resistance space. In resistance space, the process behaves like the trace of Brownian motion, meaning it cannot easily cross large gaps. In physical space, the gaps in the environment $\omega$ disappear, but their effect on the path is still visible.
• Figure 2: Simulation of $(X_t)_{t\geq 0}$ in the cases $\lambda=100$ (top row), $\lambda=500$ (middle row) and $\lambda=2000$ (bottom row), with $\rho=0.9$, $\beta=0$ and $3\cdot 10^3$ steps. Note that the resistance space in bounded from the right, where vertical lines become infinitely dense. The process in resistance space still behaves like the trace of Brownian motion, but time-changed so that it slows down as it approaches the accumulation point.
• Figure 3: Simulation of the process with random holding times in physical space (left column) and in resistance space (right column), with parameters $(\rho,\kappa)$ equal to $(\frac{17}{22},\frac{17}{18})$ (top row), $(\frac{17}{20},\frac{17}{20})$ (middle row) and $(\frac{17}{18},\frac{17}{22})$ (bottom row), and $\beta=\lambda=0$. The values are chosen such that $1/\rho+1/\kappa$ is constant, hence we expect the same spatial scaling for all three realizations. The size of the triangles is proportional to the holding time $\tau_i$ at the site.
• Figure 4: The circles on the upper line denote the sites $(\omega_i)_{i\in\mathbb Z}$ of the Poisson process, while on the lower line, the sites have been transformed by $\omega_i\mapsto R^{0,0}(\omega_0,\omega_i)$. The gray lines connect sites with their images. In principle, the random walk can jump between any sites $\omega_i$ and $\omega_j$, but we will see that the process is 'almost nearest-neighbor', in the sense that we can disregard all edges except the nearest-neighbor edges and those that help bridge a big edge (shown above). The contribution from the edges of the second type is encoded in the random variables $(\chi(i))_{i\in\mathbb Z}$. If two big edges are close, then the bridge-edges can intersect, as shown above in red and blue. However, this will only happen with vanishingly small probability.
• Figure 5: The top diagram shows a portion of the new graph for $a_n=3$ in the case $\omega_0\in\mathcal{B}_n$ (shifted vertically for clarity). Note that all paths between $\omega_{-3}$ and $\omega_4$ are disjoint, so we can compute $\widehat{R}(\omega_{-3},\omega_4)$ by the parallel law. To recover the original graph, we first 'merge' every dashed circle into a single vertex, replace the resulting parallel edges by a single edge with the appropriate resistance (red/blue edge in the bottom diagram), and then add missing edges (not shown).

Theorems & Definitions (51)

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