Long-Range Correlation of the Sine$_β$ point Process

Abstract

We study the correlations of the celebrated Sine$_β$ point process. This point process arises as the bulk scaling limit of $β$-ensembles and has a geometric description through the Brownian carousel, as shown by Valkó and Virág (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/β$ for large $β$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valkó (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $β>0$ and $k \geq 1$, significantly extending previous results limited to specific values of $β$ or $k$.

Figures (4)

• Figure 1: (color online). Trajectories of the diffusions $\alpha_6$ (blue) and $\alpha_{20}$ (red) for $\beta=1$. This realization of $\text{Sine}_1$ has two points in $[0,6]$ and 4 points in $[0,20]$. Remark that the $\alpha_\lambda$ can not go below a level $2k\pi$ once it has been crossed, and that the dynamics freeze after times much larger than $|\log\lambda|$.
• Figure 2: Simulation of the oscillations of $t\mapsto \cos(\alpha_r(t))$, $r=100, \beta=4$. At first, the dynamic is deterministic. After times of order $T_r$, randomness appears, until the system gets frozen and converges towards $1$.
• Figure 3: (color online). On the left figure, the blue path is a sample of the trajectory of $\alpha_\lambda$ started at $\pi$ under the Girsanov change of measure \ref{['eq:girsanov']} with parameters $\lambda=1$, $\beta=200$. On the right figure, the blue path is a sample of the trajectory of $\alpha_\lambda$, started in $0$ at time $T/2$ "conditioned to attain $2\pi$" (that is with the Girsanov change of measure of \ref{['eq:girsanov']} until it reaches $\pi$ and then back to the initial measure), with $\lambda=1$, $\beta=4$. In both figures, the red path is a sample of the trajectory of $\alpha_\lambda$ without any change of measure and with the same parameters and driving Brownian motion.
• Figure 4: Plot of the potential of the diffusion $R$ at a large time $t$, for $\lambda=1$.

Theorems & Definitions (29)

• Theorem 1.1: Decay of the truncated two-point correlation function
• Theorem 1.2: Decay of partially truncated $k$-point correlations
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Truncated $k$-point correlation decay
• Remark 2.1: $\beta = 2$ transition
• Theorem 2.2: Overcrowding estimate, Holcomb, Valkó, adapted from holcomb_overcrowding_2015
• proof : Proof sketch
• Lemma 2.3: Finite time approximation
• Lemma 2.4: Discretization