This work considers a random average process of \(N\) particles on a ring of length \(L\), with a distinguished tracer particle. The \(N-1\) non-tracer particles move with rate one, choose a direction (left or right) uniformly at random and moves by a distance \(\eta(x_{i\pm 1}-x_i)\), where \(x_i\) is the position of the \(i\)-th particle (numbered starting from the tracer particle), and \(\eta\) is a random number with distribution \(R(\cdot)\) which gives the fraction of the available distance by which the particle moves. Particles cannot cross, so the numbering remains consistent. The tracer particle jumps according to the same mechanism, except that the rate of jumps to the left (resp. to the right) is given by a positive number \(p\) (resp. \(q\)) instead of \(1/2\).
The authors study the stationary distribution of the gaps between particles \(g_i:= x_{i+1}-x_i\), in the particular situation where it can be written in a factorized form \(\mathbb P_{N,L}(g_1,\dots,g_N)=Z_{N,L}^{-1}\prod w_i(g_i)\mathbf{1}_{\sum g_i=L}\).
In the symmetric case \(p=q\), the authors show that if the distribution can be written in such a factorized form, it is necessarily \(Z_{N,L}(\beta)^{-1}\prod g_i^{\beta}\mathbf{1}_{\sum g_i = L}\), where \(\beta\) is a parameter that explicitly depends on the moments of \(R\). They also present sufficient conditions on \(R\) for this situation to hold, and explicit examples that satisfy their conditions.
In the asymmetric case \(p \neq q\), the tracer particle is driven, and the stationary distribution corresponds to a non-equilibrium steady state that carries a current. In this situation, rather than an exact formula, the authors study the asymptotics of the distribution of gaps as \(N, L\) go to infinity with fixed mass density \(w_0 = L/N\). They show that if the distribution behaves like \(\prod P_i(g_i, w_0) \mathbf{1}_{\sum g_i = L}\), then \(P_i\) is necessarily of the form \(P_i(g) = \frac{\beta^{\beta}}{m_i\Gamma(\beta)}\left(\frac{g}{m_i}\right)^{\beta -1}e^{-\beta g/m_i}\), where \(\Gamma\) is the usual gamma function, and \(m_i\) is the average of the gap \(g_i\), which has an explicit formula that depends on \(i, p, q\) and \(w_0\). The authors also obtain formulas for the asymptotic behavior of the correlations between the gaps.