What numbers do you get when you iteratively scale a table? Approximations of these numbers have been used since the 1930s to predict telephone traffic and in other applications. But mathematically, the exact values are extremely complicated!

Our paper identifying the structure behind these numbers:
Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m × n matrix (25 pages).
https://arxiv.org/abs/2409.02789

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Other references:

Marco Cuturi, Sinkhorn distances: lightspeed computation of optimal transportation distances, Advances in Neural Information Processing Systems 26 (2013) 2292-2300.
https://papers.nips.cc/paper_files/pa...

J. Kruithof, Telefoonverkeersrekening, De Ingenieur 52 (1937) E15-E25.
English translation by Pieter-Tjerk de Boer:
https://wwwhome.ewi.utwente.nl/~ptdeb...
See Appendix 3d.

Melvyn B. Nathanson, Alternate minimization and doubly stochastic matrices, Integers 20A (2020) Article A10 (17 pages).
https://math.colgate.edu/~integers/up...

Robert D. Putnam, Bowling Alone: The Collapse and Revival of American Community (2000) Simon & Schuster.
https://archive.org/details/bowlingal...

Richard Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, The Annals of Mathematical Statistics 35 (1964) 876-879.
https://doi.org/10.1214/aoms/1177703591

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0:00 Predicting telephone traffic
1:16 Kruithof's example
6:02 2x2 tables
8:35 3x3 tables
15:56 Rewriting the equation for 3x3 tables
22:20 Compact equation for 3x3 tables
24:47 Larger tables
27:40 Answer to Kruithof's example

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Animated with Manim. https://www.manim.community
Music by Callistio.
Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/

Web site: https://ericrowland.github.io
Twitter:   / ericrowland