Why does my walnut cake fall?
This "cake" is practically a souffle, and falls for the same reason a souffle does: it has nothing to keep it tall. It first rises because of the steam created during baking, but once the steam production is over, it deflates. Unlike real cakes, which include flour, there is nothing to create a persistent structure here. 

There is nothing to prevent a souffle from falling when it has cooled, it is like expecting a punctured baloon to stay inflated. I would suggest using a different recipe. Alternatively, you can decide that you are OK with a moist, fudgy cake without much rise. To avoid a mishappen shape from first rising too much and then falling, whip the eggs less and combine them more thoroughly with the rest of the ingredients, while still keeping some air in.