Convergence to the exact kinetics of the one-dimensional Riviera model

The Riviera model is a random sequential deposition model for house building on a lattice, in which a house cannot be built when both nearest-neighbor sites are already occupied. In one dimension, an attempted deposition is therefore rejected whenever the target site is an isolated vacancy. Despite the apparent simplicity of this rule, no exact analytical solution of the original model is currently known. We derive an infinite hierarchy of kinetic equations governing its dynamics and introduce a systematic closure scheme that can be implemented at arbitrary finite order. Solving the resulting systems order by order, we show that the corresponding kinetics converges rapidly toward the exact behavior. At high order, the method yields an exceptionally accurate estimate of the jamming density while retaining explicit analytical expressions for the full time-dependent evolution.

Pascal Viot · 2026