Inspired by the interpretation of two dimensional Yang-Mills theory on a cylinder as a random walk on the gauge group, we point out the existence of a large N transition which is the gauge theory analogue of the cutoff transition in random walks. The transition occurs in the strong coupling region, with the 't Hooft coupling scaling as alpha*log(N), at a critical value of alpha (alpha = 4 on the sphere). The two phases below and above the transition are studied in detail. The effective number

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... degrees of freedom and the free energy are found to be proportional to N^(2-alpha/2) below the transition and to vanish altogether above it. The expectation value of a Wilson loop is calculated to the leading order and found to coincide in both phases with the strong coupling value.