Large deviation function for the Eden model and universality within the one-dimensional Kardar-Parisi-Zhang class

It has been recently conjectured that for large systems, the shape of the c entral part of the large deviation function of the growth velocity would be universal for all the growth systems described by the Kardar-Parisi-Zhang equation in 1+1 dimension. One signature of this universality would be that the ratio of cumulants R-t=[[h(t)(3)](c)](2)/[[h(t)(2)](c)[h(t)(4)](c)] wo uld tend towards a universal value 0.415 17... as t tends to infinity, prov ided periodic boundary conditions rue used. This has recently been question ed by Stauffer. In this paper we summarize various numerical and analytical results supporting this conjecture, and report in particular some numerica l measurements of the ratio R-t for the Eden model.