Haagerup's phase transition at polydisc slicing (Q6612318)

Let \(\xi_1,\xi_2,\ldots\) be a sequence of independent random vectors, uniformly distributed on the unit Euclidean sphere in \(\mathbb{R}^d\), and for \(q>-(d-1)\) let \(c_d(q)\) be the maximal positive constant such that\N\[\N\left\lVert\sum_{k=1}^na_k\xi_k\right\rVert_q\geq c_d(q)\left\lVert\sum_{k=1}^na_k\xi_k\right\rVert_2\N\]\Nfor each \(n\geq1\) and all scalars \(a_1,\ldots,a_n\), where \(\lVert X\rVert_q=\mathbb{E}[|X|^q]^{1/q}\) for a random variable \(X\). Denoting\N\[\Nc_{d,2}(q)=\frac{1}{\sqrt{2}}\left(\frac{\Gamma(\frac{d}{2})\Gamma(d+q-1)}{\Gamma(\frac{d+q}{2})\Gamma(d+\frac{q}{2}-1)}\right)^{1/q}\,,\N\]\Nwhich corresponds to \(n=2\) and \(a_1=a_2=1/\sqrt{2}\), and\N\[\Nc_{d,\infty}=\sqrt{\frac{2}{d}}\left(\frac{\Gamma(\frac{d+q}{2})}{\Gamma(\frac{d}{2})}\right)^{1/q}\,,\N\]\Nwhich corresponds to the choice \(a_1=\cdots=a_n=1/\sqrt{n}\) as \(n\to\infty\), the authors show that if \(d\geq5\) and \(-(d-4)\leq q<0\) then \(c_d(q)=c_{d,\infty}(q)\). Further, in the case where \(d=4\) they also show that for \(-3<q<0\) we have\N\[\Nc_4(q)=\begin{cases} c_{4,2}(q)\,, & \text{if }-3<q\leq-2\,,\\\Nc_{4,\infty}(q)\,, & \text{if }-2\leq q<0\,. \end{cases}\N\]\NThat is, in dimension four, Haagerup's phase transition occurs at exactly \(q=2\).