Full-counting statistics of random transition-rate matrices

We study the full-counting statistics of current of large open systems through the application of random-matrix theory to transition-rate matrices. We develop a method for calculating the ensemble-averaged current-cumulant generating functions based on an expansion in terms of the inverse system size. We investigate how different symmetry properties and different counting schemes affect the results.

Figure 1

The master equation studied here can be thought of as describing the diffusion of a particle on a network where each node is coupled to every other one and a transition from node $i$ to node $j$ occurs with rate ${\Gamma }_{ij}$. Here we sketch such a network with $N=6$ nodes. The occupied node is shown as a filled circle, the unoccupied nodes as empty. To obtain the FCS, a counter is inserted between two of the nodes, which counts every time the system moves between the two nodes.

Figure 2

Probability density histograms for entries of the steady-state vector for different asymmetric ensembles with ${10}^4$ matrices. Tunnel rates were chosen randomly from (a) the exponential distribution with $\overline{\Gamma }=1$, and (b) the $\Gamma$ distribution with $\alpha =2$, $\beta =1$. The results are shifted by the mean and scaled with factors $N^{3/2}$ and $c=\overline{\Gamma }{\left[2(\overline{{\Gamma }^2}-{\overline{\Gamma }}^2)\right]}^{-1/2}$ as in the analytic distribution of Eq. (28) (also shown).

Figure 3

Probability density histograms for nondiagonal entries of the pseudoinverse for ${10}^4$ matrices from (a) symmetric and (b) asymmetric ensembles. Tunnel rates were chosen randomly from the exponential distribution with $\overline{\Gamma }=1$. The analytical approximation of Eq. (37) is also shown.

Figure 4

As Fig. 3 but with rates distributed according to the $\Gamma$ distribution with $\alpha =2$, $\beta =1$.

Figure 5

Ensemble-averaged Fano factors for both symmetric and asymmetric ensembles with tunnel rates chosen from the $\Gamma$ distribution with parameters $\alpha$ and $\beta$. We plot the scaled deviations from the Poisson value of unity. The continuous lines shows analytic results of Eqs. (49) and (50), whereas the symbols show numerical averages from ${10}^4$ matrices of size $N=50$.