Higher-rank local mixing for cusped positive representations

Let $\mathrm{\Gamma }<{\mathrm{PSL}}_2(\mathbb{R})$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho :\mathrm{\Gamma }\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen–Margulis–Sullivan measures on $\rho (\mathrm{\Gamma })\mathrm{\backslash }G\mathrm{/}M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $\rho (\mathrm{\Gamma })$, correlations of smooth compactly supported functions along $\exp (tu)$, normalized by $t^{({\mathrm{rank}}_{\mathbb{R}}G-1)\mathrm{/}2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(\rho (\mathrm{\Gamma })\mathrm{\backslash }G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $\mathrm{\Gamma }$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $\rho (\mathrm{\Gamma })$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.