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We derive "quenched" subdiffusive lower bounds for the exit time tau(n) froma box of size n for the simple random walk on the planar invasion percolationcluster. The first part of the paper is devoted to proving an almost sureanalog of H. Kesten's subdiffusivity theorem for the random walk on theincipient infinite cluster and the invasion percolation cluster using ideas ofM. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds onthe instrinsic distance in these graphs and general inequalities for reversibleMarkov chains. In the second part of the paper, we present a sharpening ofKesten's original argument, leading to an explicit almost sure lower bound fortau(n) in terms of percolation arm exponents. The methods give tau(n) \geqn^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distanceand (assuming the exact value of the backbone exponent) kappa can be taken tobe 17/384 on the hexagonal lattice.

Subdiffusivity of random walk on the 2D invasion percolation cluster