If $F(v_1,\dots,v_k)$ is a $k$-linear form on $\mathbb R^n$, the norm I want to consider is

$$ ||F|| = \sup \frac{ F(v_1,\dots, v_k)}{\prod_{i=1}^k \left|\left|v_i\right|\right|} $$

where the vector norm is the $2$-norm.

A random $k$-linear form is one given by a $k$-tensor with i.i.d. mean 0 entries.

Known result 1: A random $2$-linear form has norm $O( \sqrt{n})$ with probability approaching $1$. (This is the same as the largest singular value of a randommatrix, so it's the square root of the largest eigenvalue of a Wishart random matrix, which is close to $2\sqrt{n}$ with high probability.)

Known result 2: A random $3$-linear form has norm $O(n^{1/2+\epsilon})$ with probability approaching 1. (On an inequality of von Neumann and an application to the metric theory of tensor products)

Known result 3: Suppose there is a sequence of distributions of $2$-linear forms on $\mathbb R^n$ such that for each degree $d$, for sufficiently large $n$, the expectation of all degree $d$ polynomials of the entries is the same as for a random $2$-linear form. Then an element of that distribution has norm $O(n^{1/2+\epsilon})$ with probability approaching one. (The expected value of $tr (M M^T)^{d/2})$ for a random matrix $M$ is proportional to $n^{1+d/2}$ and it is an upper bound on the $d$th power of the norm, so we get an upper bound of the form $n^{1/2 + 1/d}$ from the $d$th moments.)

Question:

Suppose there is a sequence of distributions of $3$-linear forms on $\mathbb R^n$ such that for each degree $d$, for sufficiently large $n$, the expectation of all degree $d$ polynomials of the entries is the same as for a random $3$-linear form. Does an element of that distribution have norm $O(n^{1/2+\epsilon})$ with probability approaching one?

Observe that the proof of known result 2 relies on computing

$$ \mathbb E \left [e^{ \lambda F(v_1,v_2,v_3)} \right]$$

for random $v_1$, $v_2$, $v_3$ on the unit ball. This can be expressed in terms of moments of $F$, but it involves polynomials of arbitrarily high degree. Is this necessary?

The motivation for this question that a small moment result would allow us to prove the same inequality for a much larger class of distributions, such as those arising from algebraic geometry.

all$v_3$. This could be significantly trickier. In the genuinely random case one can use epsilon-net arguments, but these are not accessible just from bounded moment hypotheses. I'll have to think about this some more. $\endgroup$