he parameters $c_m$ are also selected randomly from a Gaussian distribution with zero mean and a variance we will specify. Under standard manipulations (Born, Markov and secular approximations), we find a master equation diagonal in the basis of eigenstates $\dot{P}_i = (\mathbb{W})_{ij} P_j$, where $P_i$ is the occupation probability of the eigenstate ${|i\rangle}$. The master operator $\mathbb{W}$ has elements $(\mathbb{W})_{ij}$ given by $$(\mathbb{W})_{ij} =  W_{j\rightarrow i}  - r_i\delta_{i,j}
\label{eq:W}$$ where the transition rates $W_{j\rightarrow i}$ are given by $$W_{j\rightarrow i} =  J(\omega_{ji})\,\, |{\langle j|}S{|i\rangle}|^2\,.
\label{eq:Wij}$$ and $J(\omega_{ji}) = 2\pi \sum_k |h_k|^2 \delta(\omega_k - \omega_{ji})$ is the spectral density of the bath with $\omega_{ji} = E_j - E_i$. We will study the case of a bath with temperature $T=\infty$ such that the rates satisfy $W_{i\rightarrow j} = W_{j\rightarrow i}$. In this work, we consider an Ohmic bath with $J(\omega) = \omega$; this choice fixes the variance of the parameters $c_m$.

At long enough times, we anticipate that all knowledge of the initial location of the exciton will be lost and the probability of finding the exciton anywhere in the lattice will be uniform in accordance with the $T=\infty$ distribution. To ascertain how long the exciton has spent in different regions of the lattice we integrate the eigenstate occupation probabilities $P_i(t)$ over time and define $O_i(t) = \int_0^t dt' P_i(t')$. We express these occupation times in the local basis as $O_m(t) = \int_0^t dt' \sum_i |{\langle m | i \rangle}|^2 P_i(t')$

![(Colour online.) Histograms of the number of jumps $k$ in time intervals $t=300/J$ for simulations (as in Fig. \[fig1\]) with $10^8$ jumps in total. Plotted is the number of time intervals in which $k$ jumps occur, $N_\text{total}(k)$, for different strengths of disorder $d$ (labelled). Shown (dashed line) is a fit to the $d=1$ points assuming a Poisson distribution.[]{data-label="fig2"}](graph_dist.png){width="