TITLE: Explain seemingly non-random figures which arise from random Poisson points with normalization
QUESTION [8 upvotes]: Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations between genes, however discussions with colleagues suggested that it more likely to be some purely mathematical phenomenon.
Small simulation confirms that very simple mathematical model will reproduce the same effect which we see in biological data (single cell RNA sequencing data to be precise).
Setup Let us generate say $1000\times n$ random numbers from Poisson distribution and put them in a matrix of the size $(1000,n).$ Normalization: divide each ROW of the matrix by its sum. (So now rows will be summed to $1$). That is all – just plot the coordinates of the first two columns on a plane.
We get figures like below (right sides – after normalization, left – before). Three plots for different $n=100,10,3$ with Poisson and one is for uniform distribution – which does NOT show similar phenomena.
Question Can one explain why for Poisson (and NOT the other tried distributions – uniform, normal, Pareto) seemingly non-random pattern arises? I mean it looks like points are aggregated into rays? What are the other distributions except Poisson will produce the same phenomenon?
Simulation code: https://www.kaggle.com/alexandervc/toy-model-for-single-cell-rna-seq-data
More figures are there.
Essentially the code is:

X = np.random.poisson(5, size = (n_samples,n) )


    v = X.sum(axis = 1)
    X2 = X / v[:,None]
    plt.scatter(X2[:,0],X2[:,1])

REPLY [3 votes]: "seemingly non-random pattern arises" because for the Poisson examples above the expected value of the random variables is 5, while for the uniform distribution it is 25.
Larger expected value makes numbers appear denser after the row normalization step, hence visually the graphs look different.
If one takes uniform distribution with the expected value 5, the picture would look similar to Poisson examples. E.g., for cases with 3 columns,
graphs look like this.

For 20 columns, we start seeing rays for both distributions as
can be seen here.