Dynamics of Liquidity Surfaces in Uniswap v3

Jimmy Risk; Shen-Ning Tung; Tai-Ho Wang

REVIEW 3 major objections 6 minor 44 references

Dynamics of Liquidity Surfaces in Uniswap v3

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that liquidity surfaces in 5 bps Uniswap v3 pools reduce to five fixed Legendre factors with AR(1)-GARCH scores, a Nelson-Siegel-like representation, and that the 30 bps pool lacks this stable basis.

desk verdict A careful first mapping of Uniswap v3 liquidity surfaces via FPCA, worth refereeing, but the rank-standardized tick grid is a load-bearing choice the paper never validates. read the letter →

arxiv 2509.05013 v1 pith:S6DP6467 submitted 2025-09-05 q-fin.TR stat.AP

Jimmy Risk , Shen-Ning Tung , Tai-Ho Wang This is my paper

classification q-fin.TRstat.AP MSC 62H2562M1091G80

keywords Uniswapv3liquiditysurfacefunctionalPCAdynamicfactormodelLegendrepolynomialsNelson-SiegelGARCHdecentralizedexchange

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper treats Uniswap v3's liquidity distribution as a surface indexed by time and by tick distance from the current price, and asks whether this surface has a parsimonious statistical structure. For high-volume 5 bps pools, it finds that a handful of principal components explain 90–95% of cross-tick variation and that these components align with a fixed Legendre polynomial basis, much as level, slope, and curvature factors describe yield curves. The factor scores evolve as AR(1) processes with GARCH-type volatility and heavy-tailed innovations. If the claim holds, a five-number summary per timestamp captures most of the liquidity profile and yields forecasts and shock responses for the whole surface.

What carries the argument

The rank-standardized log-liquidity surface y_t(x) on x∈[−1,1], where x is a relabeled tick distance from the current price and M=201 equally spaced grid points cover 100 jump ticks on each side. The decomposition y_t(x)=m(x)+Σ_{k=1}^K β_{t,k}u_k(x)+r_t(x) separates static shape from temporal dynamics. Subspace projection distance d(U1,U2)=½‖P1−P2‖²_F compares empirical PCA bases to each other and to a fixed Legendre basis, with the random-subspace distance K(1−K/M) serving as the no-alignment benchmark. Legendre polynomials P_0 through P_4 supply the fixed interpretable basis; AR(1)-GARCH models with heavy-tailed innovations describe the score dynamics.

What would settle it

Count the timestamps in the 5 bps pools where either side of the current tick has fewer than 100 jump ticks; if these are common, the grid assumption fails. More directly, rerun the rolling-window PCA on a high-volume 5 bps pool not studied here: if the subspace distance d(U_t, U_L) stays near the random-subspace benchmark K(1−K/M) or the squared score residuals show no GARCH autocorrelation, the paper's central claim of stable Legendre-aligned factors would be contradicted.

Watch

Extended reading notes

Core claim

For the 5 bps (0.05% fee) pools, ETH-USDC on Ethereum and ARB-USDC on Arbitrum, the log-liquidity surface y_t(x) admits a stable, low-dimensional factor structure: rolling-window PCA eigenmodes span nearly the same subspace as the first few Legendre polynomials, as measured by projection distances, and five factors capture 90–95% of cross-tick variance. The factor scores follow AR(1) dynamics with conditional heteroskedasticity and Student-t innovations, so the surface has a dynamic Nelson-Siegel form: a fixed, interpretable basis (level, slope, curvature, and two higher-order shape terms) whose loadings are simple time series. The 30 bps ETH-USDC pool, in contrast, shows a drifting basis an

Load-bearing premise

At every sampled time, the pool must have at least 100 distinct liquidity-range boundaries on each side of the current price tick; if any window lacks enough jumps, the equally spaced grid on which the entire surface analysis runs is undefined, and the PCA results could be an artifact of the coordinate construction.

Editorial extensions

If this is right

Five Legendre factor scores per timestamp compress the full 201-tick liquidity profile, making surface forecasting a matter of forecasting five univariate time series.
The fixed basis allows meaningful cross-pool and cross-period comparisons of level, slope, and curvature without re-estimating the factor basis each window.
GARCH structure on the scores yields time-varying uncertainty bands for the liquidity surface, useful for quantile risk assessment and LP capital planning.
The 30 bps pool's nonstationary basis indicates that fee tier shapes liquidity dynamics, so factor models valid for 5 bps pools will not transfer blindly.
The decomposition gives a direct language for stylized facts, such as the central concentration of liquidity (negative average curvature coefficient) and the flattening response to volatility shocks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Legendre alignment generalizes to other dominant 5 bps pools, the equilibrium cross-sectional liquidity shape may be effectively a smoothed bell curve: the sign pattern of average Legendre coefficients is what a concentrated unimodal profile implies, a check that does not require PCA.
The AR(1)-GARCH score dynamics give a complete one-step predictive distribution for the surface; a direct out-of-sample test would compare quantile forecasts built from the first window's score model against empirical coverages in later windows.
The stability contrast between 5 bps and 30 bps pools suggests a behavioral hypothesis the paper does not pursue: active, fee-sensitive LPs keep dominant pools' liquidity concentrated and stable, while passive LPs create the drifting basis; on-chain position data could test whether LP turnover explains basis drift.
The paper's proposed SPDE with the Jacobi operator has Jacobi polynomials as eigenfunctions, and the Legendre basis is the α=β=0 special case, so calibrating the first few eigenmodes to the estimated factor scores would produce a continuous-time model that inherits the empirical basis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is the first systematic dynamic factor study of Uniswap v3 liquidity as a functional surface, and the empirical work is mostly careful. The main claims—stable low-rank structure and Legendre alignment for 5 bps pools—are new and interesting. But the central alignment result lives on a coordinate system the paper never validates, and that is the difference between a solid empirical contribution and one that needs a major revision.

What the paper does well: it applies FPCA and rolling-window subspace distances in a sensible way, compares against a random subspace baseline, runs a broad BIC-based model selection across GARCH variants, and is honest about ETH30 not fitting the pattern. The AR(1)+GARCH+heavy-tail structure of the factor scores is convincing as a description of the scores in the chosen coordinates. The robustness checks over M, T, and sampling frequency are a genuine strength, and the writing is unusually clear about what is an assumption and what is a finding.

The soft spots are real. The stress-test note is right: Appendix B maps to jump ticks, not price ticks. The paper says “jumps typically occur at each step near the current price,” but nowhere shows this holds in the data. If jump density varies with price or over time, the equally spaced x-grid is not affine in log price, and the PCA eigenfunctions, the subspace distances, and the Legendre alignment could all be artifacts of the rank grid rather than properties of liquidity as a function of price. The robustness checks vary M, T, and sampling frequency, but not the coordinate construction itself. That is a load-bearing omission.

There is also no released data or code, and the statistical evidence is mostly point estimates: no confidence bands for the subspace distances or CPVE curves, informal ADF tests with implicit multiple comparisons, and no bootstrap anywhere. These are fixable, but they cap how strongly the empirical claims can be stated.

None of this destroys the paper. The descriptive findings are new, the methods are standard enough to be reproducible once data appear, and the failure of the 30 bps pool to show stable structure is itself a useful observation. For someone working on AMM liquidity, DeFi market microstructure, or functional factor models, this is a valuable map of a new object. I would send it to a serious referee, but I would expect the coordinate issue—either a validation that the rank grid tracks price-relative ticks, or a reframing of the claims as being about the jump-rank surface—to be addressed before publication.

Referee Report

3 major / 6 minor

Summary. The paper studies the empirical dynamics of the Uniswap v3 liquidity surface L_t(x), where x is a relative tick distance scaled to [-1,1]. Using functional PCA / dynamic factor methods on three pools (ETH-USDC 5bps on Ethereum, ETH-USDC 30bps on Ethereum, ARB-USDC 5bps on Arbitrum) over multiple windows, it reports three main findings: (1) a low-rank structure with roughly 5 principal components explaining 90–95% of the variation in the 5bps pools; (2) stability of the empirical eigenfunctions over rolling windows and close alignment with the first five Legendre polynomials, leading to a Nelson–Siegel-like fixed-basis representation; (3) temporal factor scores that are well described by AR(1) models with GARCH-type volatility and heavy-tailed (Student-t) innovations. The ETH30 pool is shown to be much less stable and not well described by this structure. Robustness checks vary the number of ticks M, window length T, and sampling frequency. The paper is explicitly descriptive rather than formal-inferential.

Significance. If correct, the paper provides the first systematic functional-data description of Uniswap v3 concentrated liquidity and a concrete, portable basis (Legendre polynomials) for dimension reduction, analogous to the Nelson–Siegel model for yield curves. The analysis is extensive and transparent: it studies multiple pools and periods, reports rolling-window behavior, compares against a random-subspace benchmark, and includes substantial robustness checks. The main empirical claims are falsifiable and quantitatively specific (e.g., rank 5 for 90–95% variance, subspace distances below random baseline). The paper also clearly separates facts found for 5bps pools from the contrasting 30bps pool. However, the central claims rest on a particular coordinate construction (jump-rank standardization, Appendix B) and on descriptive summaries without formal uncertainty quantification, so the significance is conditional on those choices being legitimate and representative.

major comments (3)

[Sections 4.2 and 4.5, Eq. (16)] The rank-standardized coordinate system is load-bearing for all three headline contributions. The x_j grid is equally spaced in the rank order of jump ticks, not in log price; it equals an affine transform of raw tick only when jump density is perfectly uniform. Uniswap v3 liquidity is a step function whose jump locations may cluster near the current price; if density varies with price or over time, the PCA eigenfunctions, their stability, and the Legendre subspace distance d(U_t,U_L) are properties of the jump-rank grid, not of the liquidity surface as a function of price. The paper explicitly notes this caveat in Appendix B ('jumps typically occur at each step near the current price') but provides no evidence that this holds for the datasets and windows used, and Appendix F varies M, T, and sampling frequency without testing the coordinate construction itself. The manuscript should (a)
[Section 4.4] The central quantitative claims (e.g., 'rank-5 structure appears adequate', 'bases are highly stable and comfortably align with the Legendre basis') are presented without any uncertainty quantification. The CPVE curves in Figure 4 are point estimates with no confidence bands, despite being computed over thousands of heavily overlapping rolling windows. Likewise, the subspace distances d(U_t,U_0) and d(U_t,U_L) in Figure 8 have no standard errors or bootstrap intervals; the comparison to a random-subspace expectation is a useful benchmark but is not a statistical test. Since the paper's central claim is precisely that the empirical eigenfunctions are close to Legendre in a stable way, the absence of error bars on these distances makes it impossible to assess whether the observed alignment is within sampling variation. The authors should provide bootstrap or analytical confidence intervals
[Section 2.2.1 and Section 4.4] The ADF tests are used informally across many factor series and windows (Tables 3, 6, 7) with a fixed p>0.1 cutoff, without multiple-comparison correction. Similarly, the BIC model-selection sweep over 45 series and 24 models yields statements like '100% preferred GARCH' and 'always preferred t/skew-t'; these are descriptive summaries but the manuscript should acknowledge the selection multiplicity and the fact that BIC differences are reported without any measure of uncertainty. This affects the stylized fact 'approximately one unit root' and the 'overwhelming' GARCH/tail claims. I do not interpret these as fatal, but they should be softened or supported by a sensitivity analysis.

minor comments (6)

[Abstract and Section 5] The text says 'We assume the x_m are equally spaced' without mentioning the rank-standardized construction; the connection to Appendix B should be made explicit at the first introduction of the grid, since the equally spaced assumption in x is not the same as equally spaced in raw ticks.
[Appendix F] The abstract claims 'the leading empirical eigenfunctions explain the majority of cross-tick variation and remain stable, aligning closely with a low-order Legendre polynomial basis.' This is a strong claim; I suggest adding 'for the 5 bps pools considered' explicitly in the abstract, and in the introduction the paper should not overstate the Nelson–Siegel analogy beyond what the evidence supports, given the caveats above.
[Appendix A] The robustness checks vary M, T, and sampling frequency, but not the rank-standardized coordinate construction. Please add a sentence explaining why the coordinate system itself is not varied, or include a robustness check with actual tick spacing.
[Throughout] The formula for the expected squared projection distance 'K(1 - K/M)' appears to be correct for the Grassmannian distance defined in Eq. (16), but the notation 'E[d_S(U_j,U)]' is ambiguous; please specify the distribution of U explicitly and cite the precise result in Meckes (2019).
[Table 3] There are several typos and duplicated words, e.g., 'where where' in Section 1, 'the the' in Section 4.4.3, and inconsistent hyphenation of 'rank-standardized' vs 'rank standardized'. A careful proofread would improve readability.
[Section 4.6.2] The ADF p-values are reported with three decimal places and some as 0.000; in the text they are described as 'suggestive' and 'not formal' but the table may give an impression of precision. Consider reporting p>0.001 as '<0.001' and noting the informal nature in the caption.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: empirical FPCA/Legendre findings are measurements with a random-subspace benchmark; the only self-citation is non-load-bearing.

full rationale

The paper's central claims are empirical descriptions of the constructed liquidity surface, not derivations from a fitted parameter. PCA is computed directly from the data, and the Legendre alignment is measured via projection distance and compared to the expected distance for random subspaces, so the alignment is not imposed by construction. No parameter is fitted to force the eigenfunctions toward Legendre polynomials, and no out-of-sample prediction is claimed from fitted inputs. The only author self-citation is [TW24], cited in background motivation for where LPs concentrate capital; it does not enter the FPCA, subspace-distance, or time-series derivations. Appendix B's rank-standardized coordinate system is a modeling choice with an acknowledged caveat: 'In practice, jumps typically occur at each step near the current price, in which case x_j is simply an affine transform of the raw tick j.' This caveat affects interpretation and generalizability—the results are stated for the jump-rank grid rather than raw tick distance—but it does not reduce any claimed result to its own input by definition. The AR(1)-GARCH findings are in-sample descriptions of the score series, not forecasts generated by a fitted model and then relabeled as predictions. Therefore no circular step is exhibited; the minor self-citation is not load-bearing.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claims rest on several modeling choices (rank-standardized coordinates, 8-hour subsampling, window size, number of factors) and on the unverified accuracy of a private data source. No new physical or economic entities are introduced. The AR-GARCH parameters are descriptive fits rather than universal constants, so they are not listed as free parameters of a derivation.

free parameters (5)

K (number of factors) = 5
Chosen because CPVE reaches 90-95% for 5bps pools; central to the low-rank claim.
M (number of relative ticks) = 201
Chosen to balance resolution and sparsity; robustness shown with M=101, M=51, M=11.
T (rolling window size) = 400
Chosen as 4.5 months; robustness checked with T=200 and T=800.
Sampling interval = 8 hours (2400 blocks ETH, 115200 ARB)
Chosen because 2-hour spacing was too sparse; 4h and 16h robustness checks in Appendix F.
Legendre basis degree = 5
Same as K; fixed basis chosen for interpretability after observing polynomial-like PCA shapes.

assumptions (5)

domain assumption The Teahouse Finance API provides an accurate record of aggregate Uniswap v3 liquidity per block.
Section 3.1: raw data sourced from an internal API; no independent verification or public data dump.
domain assumption At each time t there are at least (M-1)/2 jump ticks on each side of the current tick, so the rank-standardized grid is well-defined.
Appendix B defines the coordinate system and notes this requirement; if liquidity is sparse, the surface is not defined on the fixed grid.
ad hoc to paper PCA on the sample covariance of undifferenced, possibly nonstationary log-liquidity yields interpretable factors.
Section 2.2.1 relies on weak stationarity for population optimality; Section 3.2.1 says they use undifferenced data despite unit roots, so the factor interpretation is heuristic.
standard math BIC selection among AR-GARCH models identifies the preferred data-generating process.
Section 4.4.3 uses BIC with Kass-Raftery cutoffs; it is a model-selection heuristic, not a formal test.
domain assumption Legendre polynomials form an appropriate comparison basis for the liquidity surface on [-1,1].
Section 4.5 compares PCA subspaces to Legendre subspaces; the choice is motivated by visual similarity, not derived from the data-generating process.

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Cite this review

Pith. "Pith review of Dynamics of Liquidity Surfaces in Uniswap v3." pith.science (2026). https://pith.science/paper/S6DP6467

@misc{pith2026250905013,
  author       = {Pith},
  title        = {Pith review of: Dynamics of Liquidity Surfaces in Uniswap v3},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6DP6467}},
  note         = {Machine review of arXiv:2509.05013}
}

abstract

This paper presents a comprehensive study on the empirical dynamics of Uniswap v3 liquidity, which we model as a time-tick surface, $L_t(x)$. Using a combination of functional principal component analysis (FPCA) and dynamic factor methods, we analyze three distinct pools over multiple sample periods. Our findings offer three main contributions: a statistical characterization of automated market maker liquidity, an interpretable and portable basis for dimension reduction, and a robust analysis of liquidity dynamics using rolling window metrics. For the 5 bps pools, the leading empirical eigenfunctions explain the majority of cross-tick variation and remain stable, aligning closely with a low-order Legendre polynomial basis. This alignment provides a parsimonious and interpretable structure, similar to the dynamic Nelson-Siegel method for yield curves. The factor coefficients exhibit a time series structure well-captured by AR(1) models with clear GARCH-type heteroskedasticity and heavy-tailed innovations.

Figures

Figures reproduced from arXiv: 2509.05013 by the authors.

**Figure 1.** Aggregated Liquidity 2.1.2. Comparison with Traditional Limit Order Books. Uniswap v3 represents a significant evolution in AMM design, effectively bridging the gap between constant function market makers (CFMMs) [AC20] and traditional electronic limit order books (LOBs). While CFMMs like Uniswap v2 offer simplicity, they lack the expressiveness of LOBs, which maintain a list of outstanding buy and sell orders at… view at source ↗

**Figure 2.** Top panel: raw log-liquidity surfaces yt(x) over time t and relative tick x according to Window 2. Bottom: cross-sectional plots of yt0 (x) against x with t0 being the first time in its respective window. notably stable shapes in the mean (although there are shifts), with some flattening occurring in Window 3 for both (more exaggerated for ETH5). ETH30 has a similar shape for the first two windows (and full average)… view at source ↗

**Figure 3.** Sample mean and standard deviation functions taken over its corresponding window, plotted against relative tick x, across all datasets and windows in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (18 more)

**Figure 4.** Figure 4: Rolling-window principal component analysis (PCA) of liquidity surfaces. Top: ordered eigenvalues (log-scale) for each window of length T = 400, color-coded by the window start date tj (lighter = earlier, darker = later). Bottom: cumulative proportion of variance expla…

**Figure 5.** Figure 5: First five PCA basis functions for each dataset. Rows: dataset, columns: basis function index. Three windows are displayed per subpanel. Note that these are unique up to a sign change. • u2(x): Represents a mixed level/slope effect: liquidity generally increases as the…

**Figure 6.** Figure 6: Time series (left) and autocorrelation functions of βt,k (right, blue) for ARB5. ACF2 (orange) refers to the autocorrelation of the squared AR(1) residuals eˆ 2 t . The gray bar indicates the 95% pointwise cutoff for significance from zero autocorrelation. Each ACF equ…

**Figure 7.** Figure 7: First five Legendre polynomials: P0(x) = 1, P1(x) = x, P2(x) = 1 2 (3x 2 − 1), P3(x) = 1 2 (5x 3 − 3x), P4(x) = 1 8 (35x 4 − 30x 2 + 3) [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: displays the subspace distance (16) for K = 3, 4, 5, 6, 7 over the rolling window start date t. The Legendre coefficients were determined as in Equation (14) using Simpson’s rule. The top panel shows d(Ut , U0), the amount by which the basis has drifted since the initi…

**Figure 9.** Figure 9: Cross-sections comparing Legendre reconstructions using K = 5 versus K = 50 over all windows and ARB5 and ETH5. Each cross-section uses the first t in that window. approximation error but risks fitting noise; K = M interpolates in-sample, a typical biasvariance tradeo…

**Figure 10.** Figure 10: ETH5: effect onto the full cross-section yt(x) of a 1 standard deviation shock on the kth component, k = 1, . . . , 5, where t is the beginning of the window considered. The observations match the bulleted discussion above, as expected. Notably, a shock of k = 1 incre…

**Figure 11.** Figure 11: Legendre Fit: Time series (left) and autocorrelation functions of Legendre coefficients βt,k (right, blue) for ARB5. ACF2 (orange) refers to the autocorrelation of the squared AR(1) residuals eˆ 2 t . The gray bar indicates the 95% pointwise cutoff for significance fr…

**Figure 12.** Figure 12: Legendre Fit: Time series (left) and autocorrelation functions of Legendre coefficients βt,k (right, blue) for ETH5. ACF2 (orange) refers to the autocorrelation of the squared AR(1) residuals eˆ 2 t . The gray bar indicates the 95% pointwise cutoff for significance fr…

**Figure 13.** Figure 13: Legendre Coefficients: Heatmap of BIC scores according to heteroskedasticity and distribution assumptions. Reported is the ∆ BIC relative to the lowest (best) for that series. All use an AR(1) mean. Any value of “–" had ∆BIC > 10. functional data, we can leverage tool…

**Figure 14.** Figure 14: Time series (left) and autocorrelation functions of βt,k (right, blue) for ETH5. ACF2 (orange) refers to the autocorrelation of the squared AR(1) residuals eˆ 2 t . The gray bar indicates the 95% pointwise cutoff for significance from zero autocorrelation. Each ACF eq…

**Figure 15.** Figure 15: Time series (left) and autocorrelation functions of βt,k (right, blue) for ETH30. ACF2 (orange) refers to the autocorrelation of the squared AR(1) residuals eˆ 2 t . The gray bar indicates the 95% pointwise cutoff for significance from zero autocorrelation. Each ACF e…

**Figure 16.** Figure 16: PCA Scores: Heatmap of BIC scores according to heteroskedasticity and distribution assumptions. Reported is the ∆ BIC relative to the lowest (best) for that series. All use an AR(1) mean. Any value of “–" had ∆BIC > 10. TARCH(1,1,1) (standard-deviation recursion). σt,…

**Figure 17.** Figure 17: PCA Scores: Heatmap of BIC scores according to mean choice. Reported is the ∆ BIC relative to the lowest (best) for that series. All use an EGARCH(1,0,1) volatility and t-distributed errors. Any value of “–" had ∆BIC > 10. Innovation distributions. All are standardize…

**Figure 18.** Figure 18: ARB5: effect onto the full cross-section yt(x) of a 1 standard deviation shock on the kth component, k = 1, . . . , 5, over the three windows considered [PITH_FULL_IMAGE:figures/full_fig_p040_18.png]

**Figure 19.** Figure 19: Legendre Coefficients: Heatmap of BIC scores according to mean choice. Reported is the ∆ BIC relative to the lowest (best) for that series. All use an EGARCH(1,0,1) volatility and t-distributed errors. Any value of “–" has ∆BIC > 10. sizes, T = 200 is much rougher and…

**Figure 20.** Figure 20: Left to right: ARB5, ETH5, ETH30. Proportion of variance explained over various choices of M (the number of x’s to use centered around the current price). Analogous to bottom panel of [PITH_FULL_IMAGE:figures/full_fig_p042_20.png]

**Figure 21.** Figure 21: Subspace distances analogous to [PITH_FULL_IMAGE:figures/full_fig_p043_21.png]

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