Your existing code will clip the pixel values at 0 and 1, so all the negative values are shown as black and all values greater than 1 as white. The intensity scale is therefore very much not linear.
Instead of imshow(a), use
imshow(a, []);
The pixel values will then be mapped to intensities approximately linearly (but only approximately, because it depends on the physical response of your screen and its associated electronics). The value nearest to -infinity will be displayed as black, and the value nearest to +infinity as white, with the other values distributed through the gray shades between.
You can also specify the mapping imshow uses with imshow(a, [blackval whiteval]) - see the documentation.
By the way, I don't think randn(20,20) + sqrt(3)*randn produces the distribution you describe. Do you perhaps mean sqrt(3)*randn(20,20)?
