  As in random matrix theories, eigenvector/value distributions are important
quantities of random tensors in their applications. Recently, real
eigenvector/value distributions of Gaussian random tensors have been explicitly
computed by expressing them as partition functions of quantum field theories
with quartic interactions. This procedure to compute distributions in random
tensors is general, powerful and intuitive, because one can take advantage of
well-developed techniques and knowledge of quantum field theories. In this
paper we extend the procedure to the cases that random tensors have mean
backgrounds and eigenvector equations have random deviations. In particular, we
study in detail the case that the background is a rank-one tensor, namely, the
case of a spiked tensor. We discuss the condition under which the background
rank-one tensor has a visible peak in the eigenvector distribution. We obtain a
threshold value, which agrees with a previous result in the literature.
