Permanent linear impact paired with realistically persistent order flow makes simulated prices superdiffuse. Sato, Fujiwara and Kanazawa quantify the failure. Mean-squared displacement scales as t^(1+2delta-alpha) whenever alpha < 2delta. Set linear impact (delta = 1) and a metaorder-size exponent of alpha = 1.5, and the result is t^1.5. One-step return autocorrelation then decays as tau^(-0.5), leaving long memory in returns. The textbook Kyle specification breaks this way throughout the empirically observed range 1 < alpha < 2.
Lillo-Mike-Farmer gets a price
The Lillo-Mike-Farmer model accounts for long-range correlation in buy-sell market-order signs. M institutional traders each draw a metaorder of size L from a power law with exponent alpha > 1. They split it into unit child orders carrying the same sign and submit them with intensity lambda = 1/M. Shared signs within each metaorder make sign autocorrelation decay as tau^(-gamma), where gamma = alpha - 1. Thus alpha in (1,2) corresponds to the observed gamma in (0,1). LMF itself contains no prices. This paper supplies them.
Each child order moves the price by epsilon times the increment of I(Q) = Q^delta, with Q denoting the volume already executed in that metaorder. Choosing delta = 1/2 puts the square-root law inside a model of long-memory order flow. The authors then map the system onto a nonlinear Lévy walk. Its walker travels for a heavy-tailed random duration, while displacement grows as t^delta during the trip. The mapping yields closed-form Fourier-z and Fourier-Laplace representations of the price distribution. Fujiwara receives credit for the exact single-trader discrete-time calculation. He and Sato handle the many-body case, obtained from P_M(k,z) = [P_1(k,lambda z)]^M.
The main theorem gives linear MSD growth for alpha > 2delta and t^(1+2delta-alpha) growth for alpha < 2delta, with a log correction at the boundary. Therefore delta <= 1/2 is necessary and sufficient for diffusion for every gamma in (0,1). The exact asymptotic return autocorrelation exponent is theta = alpha - 2delta + 1. Whenever delta <= 1/2 and alpha > 1, theta > 1. Correlation is integrable, a central limit theorem applies, and the price diffuses.
Trader count leaves the exponent unchanged. The M-body MSD equals exactly M times the single-trader MSD, a result checked at M = 1, 5, 10. Two extensions preserve the same MSD formula. One inserts an exponential resting time between metaorders. The other lets impact decay exponentially to a permanent level.
There is no dataset. The evidence consists of analytics and Monte Carlo on a grid of delta in {0.25, 0.5, 0.75, 1.0}, alpha in {1.25, 1.5, 1.75}, M in {1, 5, 10}. A cited survey of the Tokyo Stock Exchange supplies the square-root universality anchor. Cited ANcerno work supplies the impact-decay anchors: a permanent-to-peak ratio near 1/3 and a 50-day decay timescale.
Why one half?
The mechanism also shows where the result stops applying. A metaorder lasting L contributes displacement of size L^delta, while durations follow a tail with exponent alpha. Step lengths consequently have tail exponent alpha/delta, and step durations retain tail exponent alpha. Variance per unit time stays finite when the displacement tail is sufficiently thin relative to the duration tail. The boundary is exactly 2delta < alpha. Concavity suppresses the rare, long-lived metaorders that would otherwise pull prices in one direction for an extended period.
The quantifier matters. The condition is necessary and sufficient for diffusion for every alpha > 1. A single observed alpha leaves room. At alpha = 1.5, superdiffusion begins only when delta > 0.75, so diffusivity alone cannot identify an exponent close to one half.
The second result closes that room. With resting periods, the generalized model produces a price-change tail whose exponent is beta = alpha/delta. Choosing alpha = 1.5 and delta = 1/2 gives beta = 3 without tuning. Raise delta to 0.75 and beta falls to 2, conflicting with the empirically observed beta of about 3. Diffusion and the tail constraint together select the square root.
Kanazawa's derivation of the tail exponent is the page I would preserve if only one page survived. It gets beta near 3 with no Kesten mechanism and no self-excitation. The authors also acknowledge that Gabaix's competing prediction delta = alpha - 1 was refuted in their own earlier Letter.
The paper supplies its own qualification. For beta > 2, the power-law tail is a transient intermediate asymptotic and eventually reverts to Gaussian behavior. The inverse-cubic panels use observation window Delta t = 10^4 against resting timescale tau_r = 10^5.
The fat tail occupies the crossover.
Permanent impact carries the theorem
Every result above assumes impact retains a nonzero permanent component. The authors state this in the main text of their remarks section, calling the presence of such a component an assumption of their models. When permanent impact is exactly zero, each metaorder's contribution disappears at long horizons and the resulting price dynamics become subdiffusive. That case belongs to an essentially different model class.
The abstract and concluding discussion present the claim without this condition: "We prove that the price dynamics are diffusive at long times under the square-root law even under predictable market-order flow". The caveat surfaces only in Sec. V C and its footnotes.
The paper cites Maitrier and Bouchaud, whose nonlinear propagator combines a zero permanent component with power-law transient decay. Their account has long-range correlations in metaorder flow offsetting decaying impact, which produces diffusive price dynamics without any permanent component. In footnote 7, the authors answer that alpha-seeking traders exist and that their informational contribution belongs in the effective dynamics anyway. On this view, a market with an exactly zero permanent component is implausible.
The paper's second extension cannot settle the dispute. Its averaged permanent impact is I(Q) times {c_D + (1-c_D) tau_D/(tau_D + tau_d)}, which remains nonzero throughout the setup, as footnote 6 observes. The impact-decay model therefore asks whether decay to a floor disrupts diffusion. It does not. The floor itself remains fixed. This account and the competing one divide over the precise quantity left unchanged by both the theorem and simulations, and the paper identifies testing it against microscopic data as future work.
For execution
Treat the result as a restriction on impact models. Calibrate a permanent linear impact term (delta = 1), then simulate LMF order flow with a metaorder-size exponent in the empirical range 1 < alpha < 2. The paper predicts superdiffusion: MSD ~ t^(1+2delta-alpha), while the return ACF exponent theta = alpha - 2delta + 1 lies below one. At alpha = 1.5, those become t^1.5 and tau^(-0.5). Any execution schedule or cost estimate built on that generator inherits the same exponents.
Concavity at or below one half restores diffusion. Pair that concavity with a metaorder exponent near 1.5 and the model also generates an inverse-cubic price-change tail, beta = alpha/delta = 3. The volatility-clustering result, C_V(tau) proportional to tau^(-xi) with xi near alpha - 1, remains a numerical conjecture. The authors explicitly say they have no proof, so I would build nothing on it.
We could not test any of this. The model requires child-order signs, metaorder identities and cumulative executed volume. Our data is OHLCV bars with no aggressor-side classification. We therefore cannot reconstruct order-splitting episodes or estimate alpha or delta, much less place a stock relative to the 2delta = alpha boundary. Trade-level prints with a signed initiator are the missing input.
A metaorder study could change my view of the fault line by measuring I_infinity net of the alpha component and finding it statistically indistinguishable from zero. Under the paper's own interpretation, such a result would shift the resolution of the paradox from concave impact to compensation between long-range correlations in metaorder flow and impact decay. The elegant Lévy-walk mapping would remain intact as mathematics while surrendering its claim on real markets.