A direct mapping between Eden growth model and directed polymers in random media

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Abstract
On the basis of preserving required symmetries and capturing the essential ingredients, the scaling properties of the frontier of Eden model clusters (1961) have been suggested by Kardar et al (1986) to be governed in the continuum limit by a Burger's equation with noise. The authors show that such a correspondence is exact by mapping explicitly the Eden model onto the problem of the conformation of a directed polymer in a random medium at zero temperature. The latter model has been shown to be itself in correspondence with Burgers' equations. In addition, this mapping allows them to have access to the distribution of noise to be included in Burgers' equation to exactly reproduce the Eden growth model. Finally, this implies that the Eden model always shows the same universal behaviour, in contrast to a recently found breakdown of universality for the directed polymer problem in a 'very disordered' medium.