Spread a handful of coins on the table. Each coin is a possible outcome: heads, tails, or something else. If we know how chaotic this set is (its entropy), which combinations of probabilities will we encounter more often? It turns out the answer depends on how much chaos we allow.

The researchers took all possible probability distributions for K outcomes and fixed the entropy, as if we were to go through all the ways to distribute a unit sum of probabilities among K bins, but only those where the degree of mixing is the same. They applied a method akin to slicing the space into equilibrium pieces and discovered a sharp transition. When entropy falls below a critical threshold (approximately log K − 1 + 0.577), almost all distributions become "condensed": one bin takes the lion's share, while the rest split the meager crumbs equally.

This sheds light on situations where complexity is limited. For example, in machine learning, a model might produce an overly confident prediction because its internal probability space spontaneously collapses to a single option. In ecology, a similar mechanism explains why a mature community often features one dominant species, with the rest forming a uniform background. Sparsity is not a whim but a natural consequence of entropic constraints in high-dimensional worlds.