concentrated liquidity
A plain constant-product pool spreads a liquidity provider's money evenly across every conceivable price from zero to infinity, even prices that will never realistically trade. That is wasteful: when ETH is around 2,000 dollars, the capital you reserved for ETH at 5 dollars or at 5 million dollars just sits idle. Concentrated liquidity, introduced by Uniswap v3 in 2021, lets a provider instead deposit only into a chosen price interval, say 1,800 to 2,200 dollars. Inside that band the position behaves like an ordinary constant-product pool but with the curve mathematically amplified, so a given amount of capital provides far deeper liquidity and earns far more fees where trading actually happens.
The trick is virtual reserves. Within a range [pa, pb], a position of liquidity L is described by x = L*(1/sqrt(P) - 1/sqrt(pb)) and y = L*(sqrt(P) - sqrt(pa)), where P is the current price. The smaller the range, the larger the effective L for the same dollars, which is the source of the capital-efficiency gains, sometimes thousands of times versus full-range provision. The whole price axis is partitioned into discrete ticks (each tick a 0.01% price step), and the pool tracks the total liquidity active in the current tick; as the price crosses a tick boundary, positions whose ranges end there are added or removed from the active liquidity.
The cost of concentration is sharper trade-offs. When the market price leaves your chosen range, your position converts entirely into the worse-performing asset and stops earning fees altogether until price returns — a much more extreme version of impermanent loss. Concentrated-liquidity provision therefore becomes an active management problem resembling market-making: you must choose ranges, rebalance as price drifts, and weigh fee income against divergence loss. It also gives rise to liquidity-provider strategies, automated vault managers, and 'just-in-time' liquidity where a sophisticated actor adds an enormous tight position for a single block to capture one large swap's fees, then removes it.
Token amounts of a concentrated position over range [pa, pb] at price P.
Concentrated liquidity does not abolish impermanent loss — it concentrates it. A tight range earns more fees but suffers larger relative loss when price moves through it, so v3 LPing only beats v2 if fee income outweighs that amplified divergence loss.