A Tensor Critical Threshold in Cubic Spherical Kernel Graphs

We identify a strong-detection transition within the critical scale for testing cubic spherical kernel graphs against $G(n,1\mathrm{/}2)$. Let $X_1,\dots ,X_n$ be independent uniform points on ${\mathbb{S}}^{d-1}$. Conditional on these points, edges are independent, with connection probability $$\frac{1+\beta Z_{3,d}(⟨X_i,X_j⟩)}{2}\mathrm{.}$$ Here $Z_{3,d}$ is the normalized cubic zonal harmonic, with multiplicity $N_{3,d}\sim d^3\mathrm{/}6$. Assume $n\to \mathrm{\infty }$, $d=d_n\to \mathrm{\infty }$, $\beta ={\beta }_n$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 1$, and $n\le d^A$, with $A<\mathrm{\infty }$ fixed. Write $r_n=n{\beta }^2\mathrm{/}(N_{3,d}d)$. On the critical scale $r_n\to r\in (0,\mathrm{\infty })$, we identify the tensor variational threshold $$r_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}}:=\underset{0<q<1}{\min }\frac{-2\{\log (1-q)+q\}}{q^6}\mathrm{.}$$ For $0<r<r_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}}$, the log-likelihood ratio under the null admits an asymptotically Gaussian expansion in the signed triangle, and the total-variation distance converges to $2\mathrm{\Phi }(r^{3\mathrm{/}2}\mathrm{/}2)-1$, where $\mathrm{\Phi }$ is the standard normal distribution function. For $r>r_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}}$, the likelihood-ratio test is strongly consistent, although the centered signed triangle still has Gaussian fluctuations. In the strict regimes $r_n\to 0$ and $r_n\to \mathrm{\infty }$, total variation tends to zero and one, respectively, even when $\beta \to 0$.