We review some correspondences between YM2 and the theory of random walks. In this spirit, we then consider YM2 in a non conventional large N limit, in which the coupling constant A is not fixed ('t Hooft scaling) but scales with a factor logN. In this regime the effective number of d.o.f. of the model is proportional to N to the power k, with k(A) less than 2, rather than to N squared. Moreover, a transition corresponding to the cutoff transition in random walks occurs when k equal to zero. This transition may be thought of as a step further in the spirit of the Douglas-Kazakov transition.