population of dots with normal distribution of pitch

Question

I want to generate a plot that shows a rectangle populated with dots, where the dot-to-dot distance (pitch) distribution is a lognormal (or a gaussian). I want to be able to change the mean dot-to-dot distance and also the sigma. I started with the line below, which generates randomly placed points.

Graphics[{White, Rectangle[{-2, -2}, {2, 2}], Black, 
  Point /@ RandomReal[{-2, 2}, {4000, 2}]}, 
 PlotRange -> {{-2, 2}, {-2, 2}}] 

The following line seem to generate a normal distribution for the dot location in the square instead of normalDistribution of the dot-to-dot distance.

Graphics[{White, Rectangle[{-2, -2}, {2, 2}], Black, 
  Point /@ RandomReal[NormalDistribution[], {4000, 2}]}, 
 PlotRange -> {{-2, 2}, {-2, 2}}]

I'd really appreciate some help. Also in the lines above, I set the number of dots to 4000, but I guess the number must depend on the Rectangle size and the mean dot to dot distance.