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A Model for Fast Mining Blockchains

This paper presents a Markovian model for blockchain mining that, in the infinite miner limit, converges to a travelling wave profile described by coupled ODEs and the Fisher-KPP equation, while deriving explicit logarithmic corrections to the mining speed for both time and finite miner populations.

Original author: Rhys Bowden, Cengiz Gazi, Jan de Gier, Peter Taylor

Published 2026-11-02

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CC BY 4.0

Original author: Rhys Bowden, Cengiz Gazi, Jan de Gier, Peter Taylor

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the digital world, a blockchain is a shared ledger that records transactions across a vast network of computers. To keep this ledger secure and up to date, independent participants called miners compete to solve complex mathematical puzzles. When a miner succeeds, they add a new block of data to the chain, which is then broadcast to the rest of the network. For the system to work, everyone must agree on the exact order of these blocks. This agreement, known as consensus, relies on the network being able to communicate new blocks faster than new blocks are being created. If the network is slow to share information, the system can handle the delay; if it is fast, the dynamics change entirely, potentially causing the network to fracture into competing versions of history.

A team of researchers has built a mathematical model to understand exactly what happens when mining is so fast that the time it takes to create a new block is roughly the same as the time it takes for that block to travel across the network. In this high-speed environment, a miner might find a new block just as another miner is receiving a different one. This creates a chaotic situation where different miners are working on different versions of the blockchain, raising the question of whether they can ever agree again. The researchers wanted to know if, in this fast-paced scenario, the network would eventually settle down into a single, unified history, or if it would remain permanently divided.

To investigate this, the authors constructed a model that simulates the behavior of a large group of miners. They imagined a scenario with an infinite number of miners to see the overall pattern of behavior, and then looked at how a finite number of miners behaves in comparison. Their analysis showed that even in this chaotic, fast-mining environment, the network does not fall apart forever. Instead, the miners eventually reach a state where they agree on the same sequence of blocks for the vast majority of the chain, but their local versions may diverge at the very tip. Specifically, the local blockchains are guaranteed to return to full consensus in finite expected time, but in the limit of many miners, they typically agree on all blocks except those that lie within a bounded distance of the head of the longest chain. This means that while the very newest blocks might be disputed for a short time, the vast majority of the blockchain remains stable and agreed upon by everyone.

The researchers discovered that the shape of this agreement follows a predictable pattern as it moves through time. They found that the "front" of the blockchain—the point where the miners are currently working—moves forward at a steady speed. However, this speed is not perfectly constant; it is slightly slower than a simple calculation would predict. The difference is a small correction that grows logarithmically with time. In simpler terms, the faster the mining, the more the network's progress is slowed down by the time it takes for information to travel, but this slowdown follows a precise mathematical rule that the team was able to derive.

Furthermore, the study examined what happens when the number of miners is limited, rather than infinite. In a real-world network, there are always a finite number of computers. The researchers found that the speed of the blockchain's progress in a finite network is also slower than in the infinite ideal case. This difference, or correction, depends on the total number of miners. Specifically, the speed slows down by an amount that is related to the logarithm of the number of miners. This means that as you add more miners to the network, the speed gets closer to the ideal infinite speed, but it does so very slowly, following a specific mathematical curve.

To make sense of these complex patterns, the team compared their results to a well-known equation used in physics and biology to describe how things spread, such as a fire moving through a forest or a species spreading across a landscape. They found that for a specific choice of parameters, the shape of the blockchain's growth in their model matched a famous, exact solution to this spreading equation. This connection allowed them to describe the shape of the blockchain's growth with a high degree of accuracy, confirming their theoretical predictions with computer simulations.

The study provides a clear picture of how fast mining blockchains behave. It proves that even when the network is overwhelmed by the speed of new blocks, it does not collapse into permanent chaos. Instead, it self-corrects, maintaining a long history of agreement while only the very newest blocks remain in flux. The researchers have quantified exactly how fast this system moves and how the number of participants affects that speed, offering a rigorous mathematical foundation for understanding the limits of speed in decentralized networks. Their work suggests that while speed introduces complexity, the fundamental structure of the blockchain is robust enough to maintain order, provided the communication between miners is not completely severed.

Technical Summary: A Model for Fast Mining Blockchains

Problem Statement
This paper investigates the dynamics of blockchain mining in a "fast mining" regime, where the time required to mine a new block is comparable to the propagation delay required to communicate that block across the network. In traditional "slow mining" models, forks (divergences in local blockchains) are typically resolved quickly because a new block is likely to be mined on one branch before the network fully synchronizes. However, in fast mining scenarios, multiple blocks may be mined simultaneously on different branches while propagation is ongoing, leading to a complex state where miners may never hold identical local blockchains. The central question is whether consensus is reached, how the distribution of blockchain heights evolves, and what the asymptotic behavior of the system is as the number of miners ( $N$ ) approaches infinity.

Methodology
The authors construct a continuous-time Markov chain (CTMC) model for a network of $N$ miners.

State Space: The state is defined by the vector of local blockchain heights for each miner. Transitions occur via two mechanisms:
- Mining: A miner discovers a new block with rate $\lambda$ , increasing their local height.
- Propagation: Miners communicate their local blockchain to others with rate $\omega$ . A receiving miner updates their local chain if the received chain is valid and has a greater height.
Finite $N$ Analysis: The authors first prove that for any finite $N$ , the system is positive recurrent, meaning consensus (all miners holding identical blockchains) is reached in finite expected time, regardless of the initial divergence.
Infinite Limit ( $N \to \infty$ ): To analyze the macroscopic behavior, the authors employ a functional law of large numbers (FLLN) based on the work of Barbour and Luczak (2008). They define an empirical distribution function $\phi_n(t)$ representing the proportion of miners with a blockchain height less than or equal to $n$ .
Asymptotic Analysis: The limiting distribution satisfies a deterministic system of coupled ordinary differential equations (ODEs). The authors analyze this system using:
- Comoving Frames: To study the traveling wave profile, they fix a level set $p \in (0,1)$ and shift the time coordinate to eliminate translation drift.
- Asymptotic Arguments: Inspired by Brunet and Derrida (1997) and Bramson (1980s), they derive corrections to the wave speed.
- Continuum Approximation: They approximate the discrete difference equations with a partial differential equation (Fisher-KPP with drift) and utilize Painlevé integrability results (Ablowitz and Zeppetella, 1979) to find explicit analytic solutions for specific parameter regimes.

Key Contributions and Results

Consensus Dynamics:
- Proposition 1: For any finite number of miners $N$ , the consensus state is positive recurrent. Even with high propagation delays, the system returns to consensus in finite expected time.
- Functional Law of Large Numbers: In the limit $N \to \infty$ , the stochastic process converges to a deterministic profile satisfying a system of ODEs (Equation 3.12):
  $\dot{\phi}_n(t) = -\lambda(\phi_n - \phi_{n-1}) - \omega\phi_n(1 - \phi_n)$
  with boundary condition $\phi_{-1} \equiv 0$ .
Convergence to a Traveling Wave:
- Theorem 12: The sequence of solutions to the ODE system converges uniformly to a non-trivial limiting profile in a comoving frame. This profile is a traveling wave with a shape defined implicitly by a boundary value problem.
- The wave speed is determined by the inter-hitting time $\tau$ of a specific level set, which is independent of the chosen level $p$ .
Wave Speed and Logarithmic Corrections:
- Minimum Speed: The traveling wave propagates with a minimum speed $v_0$ determined by the linearization of the system at the leading edge.
- Bramson Correction: For the infinite miner limit, the position of the front $X_t$ exhibits a leading-order logarithmic correction in time:
  $X_t \sim v_0 t - \frac{3}{2\gamma_0} \log t$
  where $\gamma_0$ is the decay rate of the wave tail.
- Brunet-Derrida Correction: For a finite number of miners $N$ , the traveling wave speed $v_N$ converges to the infinite limit $v_0$ with a correction proportional to the inverse square of the logarithm of $N$ :
  $v_0 - v_N \sim \frac{C}{(\log N)^2}$
  This result aligns with the Brunet-Derrida behavior observed in cut-off modified Fisher-KPP systems.
Analytic Approximation of the Profile:
- By mapping the discrete system to a continuum Fisher-KPP equation with drift, the authors identify a specific parameter regime where the system admits a Painlevé integrable traveling wave solution.
- Proposition 28: For a specific ratio of $\omega/\lambda \approx 0.00781$ , the exact analytic solution derived by Ablowitz and Zeppetella (1979) accurately matches the limiting profile of the blockchain model. Numerical simulations confirm a strong fit between this analytic solution and the numerical solution of the discrete ODEs.

Significance and Claims
The paper claims to provide a rigorous mathematical framework for understanding fast mining blockchains, moving beyond heuristic models.

It establishes that while consensus is not instantaneous in fast mining, the system remains stable and converges to a consensus state in finite expected time for any finite $N$ .
It characterizes the "fuzziness" of the blockchain head in the infinite limit as a traveling wave with a specific shape and speed.
It quantifies the finite-size effects, demonstrating that the speed of propagation in a finite network is slower than the infinite limit by a factor of $(\log N)^{-2}$ .
The work bridges discrete stochastic processes (blockchain mining) with continuous reaction-diffusion equations (Fisher-KPP), utilizing advanced asymptotic techniques (Bramson, Brunet-Derrida) and integrable systems theory (Painlevé) to derive explicit expressions for wave speed and profile shape.

The authors support these theoretical findings with numerical simulations, verifying the logarithmic corrections in time and miner count, and the accuracy of the Painlevé-based analytic approximation for the limiting profile.

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