This paper shows that graph spectral embedding using the random walk\nLaplacian produces vector representations which are completely corrected for\nnode degree. Under a generalised random dot product graph, the embedding\nprovides uniformly consistent estimates of degree-corrected latent positions,\nwith asymptotically Gaussian error. In the special case of a degree-corrected\nstochastic block model, the embedding concentrates about K distinct points,\nrepresenting communities. These can be recovered perfectly, asymptotically,\nthrough a subsequent clustering step, without spherical projection, as commonly\nrequired by algorithms based on the adjacency or normalised, symmetric\nLaplacian matrices. While the estimand does not depend on degree, the\nasymptotic variance of its estimate does -- higher degree nodes are embedded\nmore accurately than lower degree nodes. Our central limit theorem therefore\nsuggests fitting a weighted Gaussian mixture model as the subsequent clustering\nstep, for which we provide an expectation-maximisation algorithm.\n