Inter-particle gap distribution and spectral rigidity of the totally asymmetric simple exclusion process with open boundaries

We consider the one-dimensional totally asymmetric simple exclusion process (TASEP model) with open boundary conditions and present the analytical computations leading to the exact formula for distance clearance distribution, i.e. probability density for a clear distance between subsequent particles of the model. The general relation is rapidly simplified for the middle part of the one-dimensional lattice. Both the analytical formulas and their approximations are compared with the numerical representation of the TASEP model. Such a comparison is presented for particles occurring in the internal part as well as in the boundary part of the lattice. Furthermore, we introduce the pertinent estimation for the so-called spectral rigidity of the model. The results obtained are sequentially discussed within the scope of vehicular traffic theory.