Effective Resistance in Fixed-Rank External-Field Measures and Constant-Stretch Correlated Sampling on the Hypersimplex

We prove an effective-resistance bound for fixed-rank external-field measures. Let $d\ge 2$ be an integer, let $m\in \{1,\dots ,d-1\}$. Let $w\in (0,+\mathrm{\infty }{)}^d$, and let $\mathsf{S}$ be an $m$-element random subset of $[d]$ distributed according to the rank-$m$ external-field measure with weights $w$, i.e., $$\mathbb{P}(\mathsf{S}=S)=\frac{\prod _{i\in S}{w}_i}{e_m(w)},S\subseteq \{1,\dots ,d\},\mathrm{\mid }S\mathrm{\mid }=m,$$ where $$e_m(w):=\sum _{\begin{array}{c}T\subseteq \{1,\dots ,d\}\\ \mathrm{\mid }T\mathrm{\mid }=m\end{array}}\prod _{\mathrm{\ell }\in T}{w}_{\mathrm{\ell }}$$ is the $m$th elementary symmetric polynomial in $w_1,\dots ,w_d$. Let $X:=(X_1,\dots ,X_d{)}^{\mathrm{\top }}$ be its indicator vector, i.e., $$X_i=\mathbb{I}\{i\in \mathsf{S}\},i\in \{1,\dots ,d\}\mathrm{.}$$ Let $\mathrm{\Sigma }:=\mathrm{Cov}(X)$, put $v_i:={\mathrm{\Sigma }}_{ii}$ for each $i\in \{1,\dots ,d\}$, and let ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_d$ denote the standard basis of ${\mathbb{R}}^d$. Our main result is that, for every $i\mathrm{\ne }j$, $$({\mathbf{e}}_i-{\mathbf{e}}_j{)}^{\mathrm{\top }}{\mathrm{\Sigma }}^{†}({\mathbf{e}}_i-{\mathbf{e}}_j)\le \frac{1}{v_i}+\frac{1}{v_j},$$ where ${\mathrm{\Sigma }}^{†}$ is the Moore-Penrose pseudoinverse of $\mathrm{\Sigma }$. As a consequence, if $$v:=(v_1,\dots ,v_d{)}^{\mathrm{\top }},D:=\mathrm{diag}(v),V:=\sum _{i=1}^d{v}_i,$$ then, as a corollary, we obtain $$\mathrm{\Sigma }⪰\frac{1}{2}(D-\frac{v{v}^{\mathrm{\top }}}{V}),$$ which establishes a factor-two relaxation of the normalized covariance bound conjectured by Anari, Haqi, and Ma. As a further corollary, combining our theorem with the recent framework of Anari, Haqi, and Ma yields a constant-stretch guarantee for correlated sampling on the hypersimplex without relying on the still-open normalized covariance conjecture assumed in their conditional result. Our result improves the logarithmic-in-$k$ stretch bound of Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc to a constant and resolves the open question posed in their work.