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Let G be a vertex transitive graph. A study of the range of simple randomwalk on G and of its bridge is proposed. While it is expected that on a graphof polynomial growth the sizes of the range of the unrestricted random walk andof its bridge are the same in first order, this is not the case on some largergraphs such as regular trees. Of particular interest is the case when G is theCayley graph of a group. In this case we even study the range of a generalsymmetric (not necessarily simple) random walk on G. We hope that the fewexamples for which we calculate the first order behavior of the range here willhelp to discover some relation between the group structure and the behavior ofthe range. Further problems regarding bridges are presented.

On the Range of the Simple Random Walk Bridge on Groups