We propose a dynamical theory of market liquidity that predicts that theaverage supply/demand profile is V-shaped and {\it vanishes} around the currentprice. This result is generic, and only relies on mild assumptions about theorder flow and on the fact that prices are (to a first approximation)diffusive. This naturally accounts for two striking stylized facts: first,large metaorders have to be fragmented in order to be digested by the liquidityfunnel, leading to long-memory in the sign of the order flow. Second, theanomalously small local liquidity induces a breakdown of linear response and adiverging impact of small orders, explaining the "square-root" impact law, forwhich we provide additional empirical support. Finally, we test our argumentsquantitatively using a numerical model of order flow based on the same minimalingredients.
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