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the state x ( t$_{1}$ ) = x$_{1}$$_{,}$$_{2}$ of the two-level fluctuator at time t$_{1}$ . As I ( t$_{1}$ ) = V G ( t$_{1}$ ), we first concentrate on correla- tion functions of G ( t ), where statistical averages over the stochastic process (1) will be denoted by 〈·〉 . In quan- tum mechanics, it is well-known that correlation func- tions can be evaluated either in the operator or in the path-integral formalism. 25 Likewise, we have the choice to apply a stochastic path-integral approach 7,26 or to use the operator formalism. Here, we stay with the opera- tor formalism introduced in the previous section. Note that the propagator P$_{mn}$ ( t$_{2}$ − t$_{1}$ ), cf. Eq. (6), denotes the conditional probability (the transfer matrix) to find the system in state x$_{m}$ at time t$_{2}$ , given that it resided in x$_{n}$ at time t$_{1}$ , P$_{mn}$ ( t$_{2}$ − t$_{1}$ ) = 〈 x ( t$_{2}$ )= x$_{m}$ | x ( t$_{1}$ )= x$_{n}$ 〉 ; i.e., within a path-integral formulation, P ( t ) involves already an integration over all possible paths between t$_{1}$ and t$_{2}$ . 

The average conductance 〈 G ( t$_{1}$ ) 〉 is readily calculated. For a system residing in a stationary state given by 〈 x ( t$_{1}$ )= x$_{n}$ 〉 = P eq n , we obtain 

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The calculation of the conductance correlator 〈 G ( t$_{2}$ ) G ( t$_{1}$ ) 〉 is more involved. We proceed slowly in order to motivate our general mapping between the calculation of correlation functions and the evaluation of matrix products of specific matrices. In order to calculate 〈 G ( t$_{2}$ ) G ( t$_{1}$ ) 〉 , we assume first that t$_{2}$ > t$_{1}$ ; classical correlators are symmetric so that the opposite ordering of times reduces to the same quantity. Using the fact that the stochastic process is Markovian, we expand the correlation function 〈 G ( t$_{2}$ ) G ( t$_{1}$ ) 〉 = ∑ $_{m,n}$G$_{m}$ 〈 x ( t$_{2}$ )= x$_{m}$ | x ( t$_{1}$ )= x$_{n}$ 〉 G$_{n}$ 〈 x ( t$_{1}$ )= x$_{n}$ 〉 ; 24 this expansion can be seen as a transfer-matrix expansion of the correlation function. We obtain the mapping for the correlator ( t$_{2}$ > t$_{1}$ ) 

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with the diagonal matrix G$_{mn}$ = G$_{n}$δ$_{mn}$ . Introducing the “interaction representation” 

of the matrix G , the correlation function Eq. (8) can be further simplified to 

where the time-ordering operator T has been included in order to relieve the restriction t$_{2}$ > t$_{1}$ . It is easy to see that the above derivation is not restricted to the second order correlation function, but can be applied in the same 

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way to higher order correlation functions. We thus arrive at the mapping 

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where the left hand side is a correlation function for the classical stochastic process involving the two-level fluctu- ator and the right hand side is a matrix element involv- ing the matrices G$_{I}$ ( t$_{n}$ ) and the vectors | 0 〉 = P eq and 〈 0 | = (1 , 1). 

We are now in the position to calculate the generating function 

for the zero-frequency current-correlation functions (mo- ments) of a classical point contact coupled to a two- level fluctuator. The moments are obtained as the Tay- lor coefficients 〈 Q $^{n}$〉 = ( − i∂$_{λ}$ ) $^{n}$χ |$_{λ}$$_{=0}$ . Alternatively, the stochastic process can be characterized by irreducible cu- mulants which are given by the expansion coefficient of the logarithm of the characteristic function 

The characteristic function χ ( λ ) can be recast in the form 

using the mapping Eq. (11). This equation can be rewritten using the well-known mapping between the “Schr¨ odinger” and the “interaction” representation, 23 e − ( h + v ) t = e − h t T exp[ − ∫ t 0 dt $^{′}$v$_{I}$ ( t $^{′}$)] with v$_{I}$ ( t ) = e h $^{t}$v e − h $^{t}$. Here, we apply the relation in the opposite direction to arrive at an expression without the awkward time-ordering, 

This formula was derived before by Bagrets and Nazarov using a stochastic path integral formulation of the prob- lem (the Fokker-Planck Hamiltonian h and the counting field λ are denoted by ˆ L and λ in their paper $^{7}$). We believe that the present approach using transfer matri- ces and the mapping of the interaction picture onto the Schr¨ odinger picture is more transparent. Note though, that the counting field λ enters differently in their work compared to ours. Here, λ couples to the classical current I whereas Bagrets and Nazarov discuss the transport of individual (quantum) particles such that λ may only en- ter in the combination exp( iλ ) due to the quantization of charge. 

For further convenience, we subtract the average charge 〈 Q 〉 = V 〈 G 〉 t in order to obtain the reduced full counting statistics 

## B. Full Counting Statistics 

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