arxiv: 2605.06060 · v1 · submitted 2026-05-07 · 💻 cs.CE · cs.SY· eess.SY

Recognition: unknown

Arbitrage and the Stability of AMM Price Tracking

Peihao Li , Nadia Dahmani , Wenqi Cai

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:45 UTC · model grok-4.3

classification 💻 cs.CE cs.SYeess.SY

keywords Automated market makersArbitrageTracking errorGeometric ergodicityDecentralized financeLiquidityBlockchain executionPrice stability

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The pith

Arbitrage ensures geometric ergodicity of the AMM tracking error when total successful correction per block meets a threshold condition.

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

The paper establishes that the gap between an automated market maker price and an external reference price forms a stochastic process that converges geometrically to an invariant distribution whenever the aggregate arbitrage correction confirmed in each block satisfies a simple threshold. A sympathetic reader would care because this supplies a rigorous, quantitative version of the informal claim that arbitrage keeps decentralized exchange prices honest without central intervention. The authors reduce all blockchain execution details such as fees, delays, ordering, and failures to one effective per-block correction quantity and derive explicit one-step bounds that tie tracking accuracy directly to pool liquidity and execution success. They illustrate the mapping with a constant-product example and confirm consistency through simulations driven by empirical proxies extracted from realized block data.

Core claim

The tracking error between the reference price and the AMM price is modeled as a Markov process. Under the block-level correction condition on the total successful arbitrage volume realized in each block, this process is geometrically ergodic and satisfies explicit one-step bounds that connect the deviation probabilities to liquidity parameters and execution quality.

What carries the argument

The block-level correction condition, which requires that the aggregate successful arbitrage correction in each block exceed a threshold that forces contraction of the tracking-error Markov chain.

If this is right

Higher liquidity and higher rates of successful correction tighten the stationary distribution of the tracking error.
In constant-product pools the width of the no-trade band is determined explicitly by fees, fixed execution costs, and local liquidity.
The optimal corrective trade size follows directly from the same liquidity and cost parameters.
Empirical proxies built from block data can organize simulations whose comparative statics match the theoretical predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers could monitor the realized correction relative to the threshold on-chain to obtain a real-time stability margin for any given AMM.
The same reduction to a per-block correction scalar could be applied to other on-chain pricing mechanisms that rely on external arbitrage.
The explicit bounds supply a direct way to choose minimum liquidity or fee parameters that achieve a target long-run tracking precision.

Load-bearing premise

That every friction in blockchain execution can be summarized without loss by the single scalar value of total successful correction achieved in each block.

What would settle it

Collect a long sequence of on-chain blocks in which the measured total correction consistently meets or exceeds the stated threshold yet the observed price deviations show no geometric contraction or violate the derived one-step bounds.

Figures

Figures reproduced from arXiv: 2605.06060 by Nadia Dahmani, Peihao Li, Wenqi Cai.

**Figure 1.** Figure 1: Probability that a block removes at least a given share of the view at source ↗

**Figure 2.** Figure 2: Reduced simulation driven by the data-guided theorem quantities. view at source ↗

**Figure 4.** Figure 4: Custom mechanism simulation based on the constant-product view at source ↗

read the original abstract

Automated market makers (AMMs) quote prices from pool state rather than from a limit order book. AMM pools often stay close to a reference price because arbitrageurs correct profitable mispricing. A large part of decentralized finance therefore relies on a simple economic premise: once the AMM price drifts away from the reference price, arbitrage incentives push it back. This paper studies when that premise is strong enough to guarantee block-scale stability. We model the gap between the reference price and the AMM price as a stochastic tracking error, treat arbitrage as the corrective input, and place blockchain execution inside the loop through fees, discrete blocks, transaction ordering, delays, and transaction failure. The detailed execution layer is reduced to the total successful correction confirmed in each block. Under a block-level correction condition, we prove geometric ergodicity of the tracking error and obtain explicit one-step bounds that connect tracking quality to liquidity and execution quality. We also show in a constant-product example how fees, fixed execution costs, and local liquidity map into the no-trade band and the optimal corrective trade. Finally, we build empirical proxies for the theorem quantities from realized block data and use them to organize reduced and mechanism-focused simulations whose comparative statics are consistent with the theory. The contribution is to turn a basic economic intuition behind decentralized finance into a quantitative stability statement together with a tractable calibration interface.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper models AMM price deviation as a stochastic tracking error, proves geometric ergodicity under a per-block correction condition, and supplies explicit bounds plus a constant-product example with block-data simulations.

read the letter

The main takeaway is that the authors reduce blockchain execution noise to a single per-block correction total and then prove that the price tracking error converges geometrically when that total satisfies their condition, with bounds that depend on liquidity and execution parameters. This framing lets them connect tracking quality directly to pool liquidity and how well arbitrage trades actually execute. They do a solid job laying out the constant-product case explicitly, showing how fees and fixed costs create a no-trade band and how to compute the optimal corrective trade. The empirical proxies from block data and the reduced simulations are a nice touch; the comparative statics match what the theory predicts, which gives some reassurance that the model isn't completely detached from practice. The soft spot is the reliance on the block-level condition itself. It aggregates a lot of messy details like transaction ordering and failures into one number, and while they give an example mapping, it's not clear how often or how robustly this holds across different AMMs or market conditions. The simulations stay mechanism-focused, so they don't fully stress-test against real market dynamics or rare events where corrections might fail more often. This is useful for DeFi researchers and protocol designers who need a quantitative way to think about stability rather than just hoping arbitrage works. It has enough formal grounding and a clear path to calibration that it should go to peer review.

Referee Report

0 major / 3 minor

Summary. The paper models the gap between an AMM's quoted price and a reference price as a stochastic tracking error process. Arbitrage is treated as the corrective input, with blockchain execution details (fees, discrete blocks, ordering, delays, and failures) reduced to the aggregate successful correction amount confirmed in each block. Under an explicitly stated block-level correction condition, the manuscript proves geometric ergodicity of the tracking error and derives explicit one-step bounds that relate tracking quality to liquidity depth and execution parameters. A constant-product AMM example maps fees, fixed costs, and local liquidity into the no-trade band and optimal corrective trade size. Empirical proxies constructed from realized block data are then used to parameterize reduced-form and mechanism-focused simulations whose comparative statics are reported as consistent with the derived bounds.

Significance. If the block-level correction condition is satisfied in practice, the results convert the standard economic intuition that arbitrage stabilizes AMM prices into a quantitative ergodicity statement with explicit, usable bounds. The reduction of execution-layer complexity to a per-block total, the constant-product calibration example, and the data-driven simulation interface together supply a tractable framework for assessing and designing AMM mechanisms. These elements are particularly valuable for DeFi protocol analysis and liquidity provision decisions.

minor comments (3)

[Introduction] The abstract and introduction state the reduction of the execution layer to total successful correction per block; a short dedicated subsection clarifying how transaction ordering, delays, and failures are aggregated into this total would improve readability without altering the formal argument.
[Constant-product example] In the constant-product example, the mapping from fees and liquidity to the no-trade band is presented clearly, but the notation for the optimal corrective trade size should be cross-referenced to the general one-step bound derived earlier to avoid any ambiguity for readers.
[Empirical proxies and simulations] The simulation section constructs empirical proxies from block data; adding a brief table that lists the exact proxy definitions (e.g., how liquidity depth and correction amounts are estimated) would make the comparative statics easier to replicate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, which correctly identifies the modeling of the tracking error process, the block-level correction condition, the geometric ergodicity result, the explicit one-step bounds, the constant-product calibration, and the data-driven simulation interface. We appreciate the recognition of the framework's potential value for DeFi protocol analysis and liquidity provision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation conditional on explicit modeling assumption

full rationale

The paper reduces execution-layer details (fees, ordering, delays, failures) to a per-block total successful correction and states this as an upfront modeling choice. It then proves geometric ergodicity and one-step bounds under the stated block-level correction condition. The constant-product example maps parameters into the no-trade band as an illustration, and empirical proxies from block data are used only for simulation checks. No fitted inputs are renamed as predictions, no self-citation chains support the central claim, and the derivation does not reduce to its inputs by construction. The result is a standard conditional theorem with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on modeling choices that summarize complex execution into a single per-block correction quantity and on the block-level correction condition; these are domain assumptions rather than fitted numbers or new physical entities.

axioms (1)

domain assumption Block-level correction condition
Invoked to guarantee sufficient arbitrage activity for the ergodicity proof.

pith-pipeline@v0.9.0 · 5546 in / 1225 out tokens · 55458 ms · 2026-05-08T03:45:48.803927+00:00 · methodology

discussion (0)

Reference graph

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