The Sine_β point process is the bulk scaling limit of β-ensembles — random matrix models parameterized by a temperature-like parameter β. At β = 1, 2, 4 you get the classical orthogonal, unitary, and symplectic ensembles. But the Sine_β process exists for all β > 0, interpolating continuously between strong repulsion (large β) and weak repulsion (small β).
Dumaz and Malvy prove that the averaged k-point truncated correlation functions decay polynomially at large separations, with a decay exponent of order 1/β for large β. The correlations are long-range — they fall off as a power law, not exponentially — and the power depends on the temperature.
The proof uses the Brownian carousel, a stochastic differential equation model for the eigenvalue process. The carousel represents each eigenvalue as a particle on a circle driven by Brownian motion with logarithmic repulsion from all other particles. The truncated correlations measure how the joint statistics of k particles deviate from independence at large separations.
The 1/β scaling means that at high β (strong repulsion, near-crystalline order), correlations decay slowly — the particles remain correlated over long distances because the repulsion enforces rigidity. At low β (weak repulsion, gas-like), correlations decay fast because the particles barely influence each other.
The result holds for all β > 0 and all k ≥ 1, extending previous work that covered only specific β values or low-order correlations. The full β-k phase space of long-range correlations, mapped by one exponent.