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The theorem of Dekking and Host regarding tightness around the mean of firstpassage percolation on the binary tree, from the root to a boundary of a ball,is generalized to a class of graphs which includes all lattices in hyperbolicspaces and the lamplighter graph over N. This class of graphs is closed underproduct with any bounded degree graph. Few open problems and conjectures aregathered at the end.

Tightness of Fluctuations of First Passage Percolation on Some Large Graphs