Abstract: We provide a theory about the random walk on a circle with radius R. The analytical form of the probability distribution of the end to end vector after n steps ω(θ,π)and theaveraged square of end to end distance <r²> are given, Our findings are the following: whenR²》nl²,<r²>≈nl²which restores the result of the random walk in a 1Dinfinite space lattice;when R²《nl²,<r²>≈πR²/3.The general expressions for the average end to end distance<|r|> and the kth moment <r^k> are given.

Key words: Random walk, Brownian motion, Polymer chain, Chain conformation