2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146192We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.33 pages, 4 figure, LaTeX; We added several references to the general framework and techniques usedMathematical PhysicsStatistical MechanicsProbability82C22, 60K35, 15A52text