Belokon and Berezovskiy's volatility hurdle gives LPs a useful pool-level measure, with little support for trading on it. Computed from four on-chain quantities, it equals the pool's on-chain implied volatility on Uniswap's volatile fee tiers. The harder claim is that the hurdle reveals competition among liquidity providers. The paper leaves that claim unresolved.
An LP on a concentrated-liquidity DEX, such as Uniswap V3 or its forks, chooses a price range on a tick grid. Fees accrue only while price remains inside that range. Every other LP with liquidity at the active tick shares those fees pro rata. Price movements also expose the LP to arbitrage losses, measured as impermanent loss or loss-versus-rebalancing (LVR).
Thousands of LPs pursue the same fee flow. The authors model that competition as a mean-field game: each LP chooses a range width and rebalancing rule, while entry and exit determine aggregate depth. Their hourly state and position events cover six WETH/USDC pools, wrapped ether against the USDC stablecoin, on Ethereum and Base from January 2024 to July 2026. A parameter-free critical volatility follows realized volatility through the sample. The headline claim places the Ethereum 0.3% pool at its free-entry margin within sampling error.
Building the hurdle
Consider an LP whose concentrated range has a half-width of w ticks and who re-centers only after price reaches an edge. Its fee share falls as 1/w: widening the range spreads the same capital more thinly. Exiting incurs boundary impermanent loss and a rebalancing cost g. Each cycle lasts about w²h²/σ². Here h is the log tick step, about 0.006 on the 0.3% tier and 0.001 on 0.05%.
The fee calculation reduces to B1 = f·V/(√q·h·L). The inputs are the fee tier f, pool volume V, price q and liquidity L at the active tick. Some width earns more in fees than it loses at the boundary only when σ falls below the critical volatility σ_c = 2√(B1·h). The participation margin, the authors' name for σ/σ_c, is denoted s. Above one, every width loses money.
Entry gives the model its prediction. Arriving LPs increase L, lowering σ_c and raising s until the best available width earns zero. The resulting zero-profit margin s⋆ is just below one, about 0.992 with a 5bp rebalancing cost. That gap disappears as costs approach zero. For the empirical calculation, σ is realized volatility from hourly log closes; B1 averages the hourly ratio over 30-day windows. Using a ratio of window aggregates instead can bias B1 by up to a factor of two, the authors report. Anyone reconstructing the measure needs to keep that distinction.
The proofs and the empirical calculations belong to different modelling layers. Existence and uniqueness are proved for an averaged game, in which rebalancing re-centers a position at a random draw from the long-run price law. All tested numbers use the exact boundary-triggered dynamics. The authors are explicit: "The two are different games, and we prove no limit theorem that connects them." Their general existence and uniqueness theorems therefore do not establish the empirical results. The closed forms depend on a restricted stationary equilibrium for boundary-triggered provision.
An identity is the paper's cleanest result. Uniswap sets h = 2f to within a relative 5×10⁻⁵, making σ_c equal to Lambert's on-chain implied volatility. This is the volatility at which fee income matches the range's loss-versus-rebalancing. The relationship changes elsewhere: σ_c is σ_IV/√2 on the 0.01% tier and about 3.3σ_IV on Aerodrome. For the standard tiers, s is simply realized over implied.
Does Ethereum's 0.3% margin of 0.983 establish equilibrium?
The January 2024 to July 2026 sample contains six WETH/USDC pools on Ethereum and Base. Five have reported margins; SushiSwap has two orders of magnitude less liquidity. Ethereum's 0.3% pool records s = 0.983, with a 95% interval of [0.945, 1.023]. That interval comes from 2,000 block-bootstrap replicates with one-week blocks. Across ten quarters, σ_c follows realized volatility as it swings from 41% to 85% annualized. The margin stays roughly between 0.9 and 1.1. This tracking is the paper's strongest evidence.
Margins elsewhere are higher. Counting fees alone, Ethereum 0.05% reaches 1.059 [1.016, 1.103], while Base Uniswap reaches 1.05. Aerodrome has gauge-emission subsidies and a margin of 1.15. PancakeSwap records 1.18, rising to 1.44 when the calculation includes only the 67% of fees paid to LPs. The authors acknowledge that their cost channel cannot by itself explain why the two Uniswap 0.05% pools exceed one: neither has emissions or a protocol fee. This agrees with our earlier finding that passive Uniswap LPs lag mainly in Ethereum's 5bp pools.
The authors supply the identification objection themselves. Suppose all volume comes from arbitrage realigning the pool. In the fast-block limit, fees then approach the active liquidity's LVR, producing s = 1 at every depth without any LP entry or exit. They write: "The level of the margin therefore does not separate this account from free entry, and a test needs the response of liquidity to changes in volume." Their cross-tier comparison rejects only a coarser explanation, with one arbitrage trade for each tick crossed. Under that account, the 0.01% tier should rank lowest; it measures highest. The authors acknowledge that the fast-block explanation, which produces exactly one, remains viable.
The abstract claims that the 0.3% ratio is below one and that an optimal width exists. Yet the interval includes one. The paper's conclusion confines itself to saying the margin matches its free-entry value within sampling error. Its wording conflicts with the abstract's "less than one". The point estimate lies between the leading-order exclusion margin of 0.9823 and the exact margin of 0.9831. At 0.983, barely below the exact threshold, the pool opens only for capital above about $2.2M. The body's "at the margin" gives the accurate verdict.
The chain records losses
Always-on provision on the Ethereum pools lost 6 to 18% of capital annually. For the 0.3% pool, the diffusive model puts that result near zero.
Exit frequency is something the diffusion gets right. When the reset policy is replayed on Base tick paths, observed exits fall within 1% of the diffusive rate for half-widths of 8% and above, and within 10% at 4%. Bipower variation attributes about 15% of Base variance to jumps. The authors' explanation is conditional: if the model prices Ethereum exits as well as Base exits, most of the gap must come from overshoot. A jump carries price beyond the boundary, a loss the model omits. Our review of LVR against a spread-paying benchmark reached the same residual.
Cost assumptions also change the verdicts. The paper labels the 17.5bp proportional cost for the 0.3% pool an upper bound. Applying the measured Base swap share lowers it to 9.8bp. The exclusion margin then rises to 0.988, and provision opens at about $30k. At s = 0.98, minimum capital is $56k at leading order and $38k under the exact calculation, a difference the authors call an order of magnitude.
The constants also assume driftless log-price, as the authors disclose. With log-drift −θσ²/2, the margin formula's 7 becomes 7 − 12θ + 4θ². It turns negative past θ = 0.79. If price is a martingale, wide ranges pay at s = 1. The "closed at every capital" verdicts for the four pools above one thus depend on the drift choice.
Screening pools
The authors used the trailing margin to gate provision across four Base pools over 7,497 hours, from September 2024 to July 2025. It kept the strategy invested for 2.5 to 12% of hours and "still does not turn a profit." We did not find a return or Sharpe figure for this test.
We could not test it ourselves.
Calculating σ_c requires active-tick liquidity and volume specific to the pool. Running the policy also requires tick crossings in their actual order. Ordinary crypto OHLCV contains none of those inputs.
The measure survives as a monitor of realized over implied volatility, calculated per pool from public state. Under the model, a fee-only LP rebalancing at the boundary loses money at any capital once the pool exceeds its exclusion margin, at most 0.9944 in this sample. To turn that monitor into evidence of free entry requires a measured response of active-tick liquidity to volume shocks. The authors themselves identify that test as the missing evidence.