  We prove that a random group in the triangular density model has, for density
larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine
isometric, and more generally $(2-2\epsilon)^{1/2p}$-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
$L^p$-fixed point properties hold and the conformal dimension of the boundary.
  In the Gromov density model, we prove that for every $p_0 \in [2, \infty)$
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
$L^p$-spaces that are $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz, and this for
every $p\in [2,p_0]$.
  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 Szemeredi's
approach to the 2-Laplacian. These in turn lead to fixed point properties using
arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results
for Kazhdan's Property (T) established by Zuk and Ballmann-Swiatkowski.
