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As in random matrix theories, eigenvector/value distributions are importantquantities of random tensors in their applications. Recently, realeigenvector/value distributions of Gaussian random tensors have been explicitlycomputed by expressing them as partition functions of quantum field theorieswith quartic interactions. This procedure to compute distributions in randomtensors is general, powerful and intuitive, because one can take advantage ofwell-developed techniques and knowledge of quantum field theories. In thispaper we extend the procedure to the cases that random tensors have meanbackgrounds and eigenvector equations have random deviations. In particular, westudy in detail the case that the background is a rank-one tensor, namely, thecase of a spiked tensor. We discuss the condition under which the backgroundrank-one tensor has a visible peak in the eigenvector distribution. We obtain athreshold value, which agrees with a previous result in the literature.

Real eigenvector distributions of random tensors with backgrounds and random deviations