How Does the Concept of “K” Being a Constant Affect the Liquidity Depth near the Current Price?

In the constant product formula (x y = k), the constant $k$ determines the overall size of the pool. However, the shape of the hyperbolic curve means that liquidity is spread thinly across all possible prices, from zero to infinity.

As the price moves away from the 1:1 ratio, the curve becomes flatter, meaning larger trades are required to move the price, which suggests deeper liquidity at those extreme points. Conversely, the curve is steepest near the center, meaning the price changes most easily with smaller trades, leading to lower liquidity depth around the current price for large trades.