We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on Lp-spaces (affine isometric, and more generally (2−2ϵ)1/2p-uniformly Lipschitz) with p varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal p for which Lp-fixed point properties hold and the conformal dimension of the boundary.In the Gromov density model, we prove that for every p0∈[2,∞) for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on Lp-spaces that are (2−2ϵ)1/2p-uniformly Lipschitz, and this for every p∈[2,p0].To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemerédi's approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to Lp-spaces previous results for Kazhdan's Property (T) established by Żuk and Ballmann–Świa̧tkowski.