The KPZ fixed point

An explicit Fredholm determinant formula is derived for the multipoint distribution of height function totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method by solving biorthogonal ensemble/non-intersecting path representation found [Sas05; BFPS07]. resulting kernel involves transition probabilities a random walk forced to hit curve defined data. In KPZ 1:2:3 scaling limit leads in transparent way formula, terms analogous kernels based on Brownian motion, invariant Markov at centre universality class. readily reproduces known special self-similar solutions such as Airy$_1$ and Airy$_2$ processes. takes values real valued functions which look locally like H\"older $1/3-$ time. Both fixed point TASEP are shown be stochastic integrable systems sense that time evolution their can linearized through new scattering transform its discrete analogue.