Compute expected number of points in [0,1] of the sine point process

Question

The Sine process is the determinantal point process with kernel $K(x,y)=\frac{\sin \pi(x-y)}{\pi(x-y)}$ when $x\neq y$ and $1$ when $x=y$.

I am wondering how to compute the expected number of points of the Sine process on the interval $[0,1]$. We have that

$$\rho(x_1,\dots,x_n)=\det K(x_i,x_j) \quad \forall i,j \in [n]$$

where $\rho$ is the factorial moment density of the determinantal point process which describes the factorial moment measure. The first factorial moment measure of a DPP gives then the expected number of points in a borel set.

$$M(B)=\mathbb{E}(N(B))=\int_B \rho(x)dx= \int_B \det K(x,x) dx=\int_B K(x,x) dx=\int_B 1 dx=m(B)$$

So when $B=[0,1]$, does this mean that the expected value of the number of points in $[0,1]$ of the sine process is simply one point?

Answer 1

It turns out that $\mathbb{E}[M(B)]=\text{Leb}(B)$ for any measurable $B\subset \mathbb{R}$ so my computation above was indeed correct.

To quote a paper by Pierre-Loic Meliot

First, since $K^{\text{sine}}(x,x)=1$ for every $x\in \mathbb{R}$, we have $\mathbb{E}[M(B)]=\text{Leb}(B)$ for any measurable subset $B\subset \mathbb{R}$. This is because we have rescaled the random point process $M_N$ in order to have an expected spacing of points equal to $1$.