A Mathematical Theory of Payment Channel Networks

renepickhardt · January 16, 2026, 12:49am

Dear fellow lightning network developers

I’ve posted a new paper on arXiv that formalizes several long-standing observations about payment channel networks (in particular with the lightning network in mind) under a single geometric framework:

https://arxiv.org/pdf/2601.04835

Many of the individual phenomena discussed in the paper are familiar to practitioners:

channel depletion,
capital inefficiency of two-party channels,
the benefits of channel factories,
and the idea that feasibility rather than routing is the real bottleneck.

The goal of this work was not to rediscover these effects, but to explain why they are structurally true and how they are connected. A key outcome of the paper is the perspective that payments should be analyzed through the set of feasible wealth distributions rather than individual paths. Liquidity states differ only by circulations within fibers over a polytope. A payment is feasible if and only if the resulting wealth vector remains inside that polytope and offchain rebalancing does not change this (as was noted by others in 2018 on lightning-dev). This leads to:

a simple throughput law S=c/r linking the supported off-chain payment bandwidth S to the rate r of infeasible payment attempts and onchain transaction bandwidth c
a cut-based characterization of feasibility,
a formal explanation of why multi-party channels (coinpools / channel factories) are structurally more capital-efficient,
and a geometric explanation of why linear asymmetric fees generically lead to channel depletion.

These insights directly motivated the recent Delving Bitcoin article, which explores Ark as a channel factory and its implications for payment feasibility:

The paper also analyzes mitigation strategies for depletion such as symmetric fees, convex/tiered fee schedules, and coordinated off-chain replenishment while explaining why some of these are difficult to deploy under today’s source-routing model.

For readers interested in simulations and executable intuition, much of the reasoning was developed alongside code and notebooks, which are collected here:

github.com/renepickhardt/Lightning-Network-Limitations

README.md

main

# Lightning-Network-Limitations

This repository is a collection of Notebooks and papers that investigate the technical limitations that exist in Bitcoin's Lightning Network Protocol with respect to its ability to scale payments. 
A lot of this is work in progress and many models depend on assumptions that can be justifiably questioned. 
Thus I encourage you to be cautious in case you use these result to draw conclusions. 

## A Mathematical theory of payment channel networks
The main document that this respository contains is a research paper titled: [a mathematical theory of payment channel networks](https://github.com/renepickhardt/Lightning-Network-Limitations/blob/paper/Limits%20of%20two%20party%20channels/paper/a%20mathematical%20theory%20of%20payment%20channel%20networks.pdf). Its contributions are:

* Providing a model and [Integer Linear Program](https://en.wikipedia.org/wiki/Integer_programming) for feasible wealth distributions of users on the lightning network given a channel graph.
* A general formular that estimates the upper bound of payment bandwidth for LN payments (currently section 4.4. of the paper) This was also [discussed during the 2024 Lighting Network Developer summit](https://delvingbitcoin.org/t/ln-summit-2024-notes-summary-commentary/1198) 
* A method to estimate how likely it is that payments are infeasible. This was also [discussed on delving bitcoin](https://delvingbitcoin.org/t/estimating-likelihood-for-lightning-payments-to-be-in-feasible/973) and in the [bitcoin optech newsletter](https://bitcoinops.org/en/newsletters/2024/06/28/).
* A proof that the number of circulations on the liquidity graph is equivalent to the number of equivalent states on the polytope of liquidity states given that two states are equivalent if they yield the same wealth distribution
* The conclusions that two party channels put (potentially too) strong constraints to the ability of liquidity to flow between peers of the network.
* **Not included in the paper yet but [available as a study](https://github.com/renepickhardt/Lightning-Network-Limitations/blob/main/Limits%20of%20two%20party%20channels/A%20Geometric%20Approach%20for%20Optimal%20Channel%20Replanishment.ipynb)**: A constructive result applying the method to optimal channel replanishment (aka circular offchain rebalancing).

## Notebooks & Studies
* While circular offchain rebalancing does not change the feasibility of payments it may positively affect the reliability of feasibile payments. A [geometric method for optimal channel replanishment](https://github.com/renepickhardt/Lightning-Network-Limitations/blob/main/Limits%20of%20two%20party%20channels/A%20Geometric%20Approach%20for%20Optimal%20Channel%20Replanishment.ipynb) is available and [was discussed on delving bitcoin](https://delvingbitcoin.org/t/research-update-a-geometric-approach-for-optimal-channel-rebalancing/1768).
* We investigate how [economic rational behaviour of nodes relates to the distribution of liquidity](https://github.com/renepickhardt/Lightning-Network-Limitations/blob/sidiropoulos_counter_example/Limits%20of%20two%20party%20channels/Estimating%20Liquidity%20State%20given%20Fees%20and%20Network%20Topologies.ipynb). This notebook in particular states that for any lightning network channel graph besides a spanning tree all channels are expected to be depleted. This goes hand and in hand with [prior research](https://github.com/gr-g/ln-steady-state-model/issues/1) (which operated under different assumptions) and seems to be a rather strong argument for the emergence of hub and spoke models (as one can understand the [results from other researchers](https://pubsonline.informs.org/doi/10.1287/mnsc.2023.03872))
* A follow up studies if the [selfish sending behavior of nodes leads to the same global optimization problem](https://github.com/renepickhardt/Lightning-Network-Limitations/blob/sidiropoulos_counter_example/Limits%20of%20two%20party%20channels/Selfish%20Sending%20yields%20Almost%20solution%20to%20Global%20Fee%20Potential%20Optimization.ipynb). The provided evidence seems to give justification to study the above global optimization problem.

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Feedback and Questions are very welcome, especially on how these results should influence protocol design and whether multi-party channels should primarily serve as channel factories or as long-lived payment venues.

With kind regards Rene Pickhardt

anon · January 16, 2026, 2:42pm

It is finished.

John 19:30

Now, please, enjoy your sabbatical.