The crux of the argument lies in the difference between the distribution of individual liquidity amounts across channels and the overall distribution of node balances resulting from the aggregation of these liquidity amounts.

By drawing a parallel with rolling two dice, where each die has an equal chance of landing on a number between 1 and 6, it illustrates how despite the uniform probability of each individual outcome, the sum of two dice does not distribute uniformly across the possible outcomes (ranging from 2 to 12). This analogy effectively communicates that even if the liquidity in each channel were to be uniformly distributed, the resulting node balance – being a sum of multiple such uniformly distributed quantities – would not itself follow a uniform distribution.

This distinction highlights the complexity in modeling network behaviors and challenges the simplification that might come from assuming uniformity in node balances based purely on the uniform distribution of channel balances. It underscores the importance of considering the aggregate effects of distributed elements when analyzing network state distributions.