# How to find the joint distribution in embedded space

Euclidean Embedding of Co-occurr ence Data under Section 2, the problem formulation is given. In the paper link, the Authors find the distribution for lower embedding for the squared distances between 2 points : $d^2_{x,y} = ||\psi(x) - \xi(y)||^2$

I am having difficulty in understanding why the probability is considered exponential. In this answer What is the PDF when X and Y are non standard distribution is different. In my case, I need to find the pdf of the square root of the distances of the error points X,Y in higher embedding and they can be correlated. How do I begin? I am new to statistics and have limited knowledge in joint distribution and having a tough time in understanding these complicated joint distribution. Please help.