Two liquidity models in Itkin's paper price a unit-size metaorder within 1.6% of each other at every tested duration. Give them an order ten times larger at half the duration, and impact differs by about 9% (1.23 against 1.13 p0). After a prior order five times larger, the peak effect on the next order is 13.24% against 7.22%. A desk has more reason to care about that split than the square-root headline, whose limits the author spells out himself.
The pool and its memory
Submitted flow moves the log-price against a counterflow, with the difference divided by constant displayed depth. Displacement uses the price that would have prevailed without the order as its reference. The reference carries the same noise as the price, so ordinary volatility never wakes the counterparties.
Those counterparties are latent traders with thresholds. They stay silent until displacement crosses a threshold, then trade against the move at a rate linear in the excess. With an exponential distribution of thresholds, their aggregate counterflow is quadratic near zero and linear at large displacement. Same-side trading depletes the pool that supplies it. Pool intensity follows a logistic map of a latent state and retains a floor of 0.3. That floor cannot be depleted. The latent state follows a generalized Langevin equation, a mean-reverting process with memory. The memory kernels sum exponentials, with two intrinsic modes and two order-flow modes in the baseline, making the Markovian lift exact.
The simulations compare four variants: a Kyle linear benchmark, a fresh pool that never depletes, a single-mode pool and the full multi-mode pool. They use dimensionless units, 2048 paths, a 0.01 time step and 81 order sizes across eight decades. There is no market data. Calibration belongs to a companion paper.
How much of the square root survives?
For a fresh pool with purely quadratic counterflow, impact follows a closed tanh form. A small order finishes before counterflow builds, leaving impact linear in size. At large size, counterflow approaches the order rate and impact grows with the square root. The exponential threshold density changes the far end: once displacement grows large, counterflow becomes linear again. The square-root range lies between two linear limits.
Remark 1 gives the exponent as 1/(k+n+1). Here k is the threshold density's order at zero, and n is the power of individual demand in the excess. The result requires k+n+1 > 1 and the overlap conditions. An exponent of one half requires k + n = 1. The paper obtains it with density positive at zero (k = 0) and linear demand (n = 1). Itkin says the exponent is not fitted to an impact curve. He also says it depends on those behavioral assumptions. The square-root explanation therefore rests on how well linear excess demand describes real counterparties; the mechanism produces the exponent implied by its demand curve.
Depletion makes the range shorter. At unit duration, the fresh pool has 2.80 decades of order sizes whose local exponent falls within 0.1 of one half. The depleted pools have 1.00 to 1.10. At unit size and unit duration (T = τ0), the local exponent is 0.51 for the fresh pool, 0.59 for the single-mode pool and 0.61 for the full pool. The last figure lies just above the paper's [0.4, 0.6] band.
The floor matters for concavity. Set it to zero in noise-free runs, and the largest local exponent reaches 1.86 at unit duration and 3.07 at 20: impact turns convex. With the floor at 0.3, the exponent stays at or below 1.0 for durations of 0.3 and up. The paper reports the comparison and says the floor "does not by itself guarantee concavity."
Duration remains in the price
At unit size, fresh-pool impact falls 18-fold as duration runs from 0.1 to 30. It goes from 0.465 at duration 0.1 to 0.0259 at duration 30. The simulations hold the detection horizon fixed; within the square-root range, impact at fixed size consequently falls roughly as T^-1/2.
Itkin proposes two variants that remove T from the law. One makes thresholds depend on order duration, and the other uses noise accumulated since onset. Both narrow the band to 3.80 decades at duration 10 against 4.80. They also ask counterparties to know the duration or the onset. The paper acknowledges that neither convention reproduces both square-root laws distinguished by Durin, Rosenbaum and Szymanski.
What does the next order inherit?
The schedule experiment gives an early warning about terminal impact as a scoreboard. Front-loading has the lowest terminal impact of four schedules, 0.0792, yet its execution-weighted displacement of 0.1487 is second highest. Flat execution posts the lowest execution-weighted displacement, 0.1352, despite twice the terminal impact.
The inherited-state test puts a unit probe immediately after a unit order. Residual displacement reduces probe impact by 12.3%. Depletion adds back 7.5%, leaving a net minus 4.8% for the full pool. Itkin calls that split an operational comparison across models; it does not separate the mechanisms exactly. Watch only the net sign and the pool can look undisturbed, as the paper points out.
Larger prior orders expose the difference between spectra. Across three spectra, the spread in peak history effects is 0.90 points after a unit prior order, 6.90 after 3x and 9.41 after 5x. Standard errors are at most 0.06 points. A broad spectrum halves its pool effect sooner than the single-mode pool, in 1.74 τ0 against 2.15 τ0; the baseline GLE takes 2.05 τ0. Yet at a gap of 50 τ0, the broad spectrum leaves a larger residual: 0.10% against 0.02% for both the single-mode and baseline pools. The paper cannot resolve the peak gap locations.
This is also the paper's stated point. Its abstract says spectra with similar single-order impacts can behave differently after substantial prior trading. Instantaneous drift "constrains only the product of pool intensity and threshold response." A single-order impact curve can therefore leave the pool dynamics unidentified. Heavy prior trading makes their memory visible in conditional impact. The conclusion is careful about the limit: these comparisons do not establish that multi-order observations identify the spectrum uniquely. The paper maps size scaling over eight decades, and the evidence remains simulated.
A theorem about the chosen price model
Under the log-price cost functional and constant depth, round-trip costs are provably nonnegative regardless of memory. Itkin notes that state-dependent depth introduces a term that can be negative, which is why he freezes depth. He also acknowledges that the balanced reference produces no permanent impact, contrary to fair-pricing arguments. None of the figures above is an execution result.
We could not test the model ourselves. Our data consists of minute OHLCV bars. It lacks signed child-order flow, quote midprices and displayed depth; the counterfactual reference is unobservable in any feed. Testing would require parent-tagged metaorders with repeated same-side executions, allowing conditional probe impact to be compared with isolated orders. Bucci et al., whom the paper cites, document the crossover empirically and need at least two liquidity time scales to fit a latent-order-book theory to it. If spectra matched on single-order impact separate as prior size grows, as they do here from 0.90 to 9.41 points, the memory spectrum earns its parameters. Even that separation, by the paper's own account, would not identify the spectrum uniquely.