# Concentrated Liquidity

To address the [lazy liquidity problem](https://docs.gnoswap.io/core-concepts/amm/problem-lazy-liquidity), GnoSwap leverages a novel architecture called **Concentrated Liquidity Pools (CLPs).** CLPs allow liquidity providers to set a minimum and maximum price range, in which their liquidity will be active. This ensures that liquidity is concentrated around a specific price range, improving capital efficiency and reducing slippage for traders.

As liquidity in CLPs only requires each amount of ***x*** and ***y*** within the designated range ***(Pa, Pb)***, its behavior can be plotted into a graph as the following:

<figure><img src="https://1737588250-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FzhHIRYhJhsVkjaycRq4b%2Fuploads%2FkwmIwBxAbJ0bmnudDwxM%2Fc%20curve.png?alt=media&#x26;token=b334c081-7ed8-4a5e-b197-ff6d5d3517bc" alt=""><figcaption></figcaption></figure>

The **real reserve curve** is a parallel translation of the **virtual reserve curve** by the number of x tokens in price b (***xb***) towards the y-axis and the number of y tokens in price a (***ya***) towards the x-axis, giving the following equation:

$$
k = (x+x\_b)(y+y\_a)
$$

Although the real reserve curve represents the actual amount of tokens deposited by a liquidity provider, the liquidity pool must track two values to compute the virtual reserve curve that the pool behaves as: liquidity ***L*** and the square root of price ***√P***.

We define ***L*** as ***√k***, a geometric mean of the quantity of two tokens in a pool, and also a constant that reflects the liquidity of the pool. We can solve for ***L*** using ***k = xy***:

$$
L = \sqrt{k} = \sqrt{xy}
$$

Similarly, we can solve for ***√P*** by simply square-rooting the price equation:

$$
\sqrt{P} = \sqrt{y/x}
$$

Using the above formulas, we can compute the virtual reserves by solving for ***x*** and ***y*** respectively:

Solving for ***x***:

$$
\sqrt{x}=\sqrt{y/P}
$$

$$
x=\sqrt{xy}/\sqrt{P}=L/\sqrt{P}
$$

Solving for ***y***:

$$
\sqrt{y}=\sqrt{Px}
$$

$$
y=\sqrt{xy}*\sqrt{P}=L*\sqrt{P}
$$

If we substitute these values for ***xb*** and ***ya*** for the initial formula, we arrive at **the concentrated liquidity formula** (the real reserve curve):

$$
k = L^2 = (x+L/\sqrt{P\_b})(y+L\sqrt{P\_a})
$$

### Output Calculation

Based on the formulas derived in the previous section, we can calculate the output of a swap.

When trading y for x (calculating ***∆x*** with ***∆y***), we first calculate **∆*****√P*** using the first equation, then use the value to calculate ***∆x:***

$$
∆√P = ∆y /L
$$

$$
∆x=∆(1/\sqrt{P})\*L
$$

Similarly, we can also calculate ***∆y*** when we trade x for y:

$$
∆(1/\sqrt{P})=∆x/L
$$

$$
∆y=∆\sqrt{P}\*L
$$

### Price Tick

Since the price range of each LP Token in a pool is unique, we need a mechanism to uniformly divide the price units of a pool to merge the liquidity of tokens provided at the same price within a single liquidity pool. The boundaries that partition the range are called **ticks**. The default tick spacing for pools is 1 basis point, meaning that liquidity is divided into intervals of 0.01%.

The pricing mechanism and the fee distribution mechanism remain the same as regular CPMM pools within a single tick. After a tick's liquidity is drained, the remaining input rolls over to the following tick to be swapped at a different rate.

Price at tick ***i*** can be expressed as the following formula:

$$
p(i)=1.0001^i
$$

Since prices are stored as a square root in GnoSwap's contracts, we use the following formula to calculate the square root of the price at tick ***i***:

$$
\sqrt{p}(i)=1.0001^{i/2}
$$

As the logarithmic calculation to derive ***i*** from a select price doesn't always end up as an integer, once a user inputs a value as a price, the contract automatically finds the nearest tick out of lower ones. For example, the exact value of the price at a tick index of 50,000, can be calculated as:

$$
1.0001^{50,000} = 148.376062923
$$

As the next highest tick (50,001) would result in a value of 148.383481541, a lower price range of 148.37 would translate into a lower tick of 50,000.