The Krawtchouk ensemble is defined by a weight: $w(x) = \binom{K}{x}p^x q^{K-x} $ and in fact it comes from a conditioned random walk on $\mathbb{Z}^N$. It is a probability measure on the set $\{ 0, 1, \dots, K\}^N $. Each sequence has probability:


$$ \mathbb{P}[h] = \frac{1}{Z} \prod\_{i < j} (h\_i - h\_j)^2 \prod\_{j=1}^N \binom{K}{h\_j} \frac{1}{2^K}$$


where $Z$ is a number such that the sum over all sequences is $\Sigma=1$.


What are the correlations of this probability distribution? Is it safe to assume this is a [determinantal process](https://terrytao.wordpress.com/2009/08/23/determinantal-processes/)? In that case it would be very easy to write down the correlations... E.g.


$$ \mathbb{P}[h\_1, h\_3] = \frac{1}{Z} (h\_3 - h\_1)^2 \times \binom{K}{h\_1} \frac{1}{2^K} \times \binom{K}{h\_3} \frac{1}{2^K}$$


However, I haven't shown it has a Kernel. Certainly the Gaussian Unitary Ensemble (GUE) *is* determinantal. The distribution of a random Hermitian matrix $N \times N$ is:


$$ \mathbb{P}(\lambda) = \prod\_{i < j} (\lambda\_i - \lambda\_j)^2 e^{- \sum\_1^N \lambda\_i^2}$$


and these probabilities can all be written as determinants because there is a Kernel:


$$K\_N(x,y) = \sum\_{i=0}^N \phi\_i (x) \, \phi\_i(y) \, e^{-(x^2 + y^2)/4} $$


This looks anwful lot like the ensemble I've just written, because of the [DeMoivre-Laplace limit theorem](https://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem)
$$ \binom{n}{k} \frac{1}{2^n} \simeq \sqrt{ \frac{2}{\pi n }} \exp\left[- \frac{(k - \frac{1}{2}n)^2}{\frac{1}{2}n} \right] $$ 
so mainly, I am asking if there's similar Kernel for the Krawtchouk distribution above.



Is the Krawtchouk ensemble a determinantal process?
The short answer is "yes, it is a determinantal process". For more details, see Johansson's paper
<https://arxiv.org/pdf/math/9906120.pdf> (which deals with the edge).


